3.1. Discussion

In this section we discuss potential extensions of Theorem 2.1 to settings satisfying weaker assumptions or involving related MCMC chains.

3.1.4. Gaussian targets in high dimensions.

Let $\pi_0$ denote a Gaussian target on ${\mathbb{R}}^d$ with mean $\mu$ and covariance matrix $\operatorname{diag}(\sigma^2_{11},\ldots,\sigma^2_{dd})$ . Inspired by [Reference Roberts and Rosenthal29, Theorem 5] and the proof of Theorem 2.1, a good control variate for the ergodic average estimator for $\pi_0(f)$ takes the form $d(P_d\skew3\tilde f-\skew3\tilde f)$ , where $\skew3\tilde f$ solves the ODE

\[\skew3\tilde{f}''+ ((\partial/\partial x^d_1)\log(\pi_0))\skew3\tilde f'=2/h_0(l)\cdot(\pi_0(f)-f),\]

with $h_0(l)\,:\!=2l^2\Phi({-}l\sqrt{J_0}/2)$ and $J_0\,:\!=1/d\cdot \sum_{j=2}^d1/\sigma^2_{jj}$ . In the case of the mean, $f(x^d_1)=x^d_1$ , we can solve the ODE explicitly: $\skew3\tilde f(x^d_1)=2\sigma^2_{11}/h(l)\cdot x^d_1$ .

If $\pi_0$ has a general non-degenerate covariance matrix $\Sigma$ , we have an ODE analogous to that above for each eigen-direction of $\Sigma$ . The control variate for the mean of the first coordinate, $f(x^d_1)=x^d_1$ , is then a linear combination of the control variates for the means in eigen-directions. Specifically, $\skew3\tilde f \colon {\mathbb{R}}^d\to{\mathbb{R}}$ in the estimator analogous to that of (1.1) (see the numerical example for $h=0$ in Section 3.2.2) takes the form

(3.1) \begin{equation} \skew3\tilde f(\mathbf{x}^d)=2/h_0(l) \cdot \sum_{j=1}^dx^d_j\Sigma_{j1},\quad \text{with } h_0(l)=2l^2\Phi({-}l\sqrt{J_0}/2) \text{ and } J_0=1/d\cdot \text{Tr}((\Sigma^{-1})_{2:d,2:d}). \end{equation}

Note that $\skew3\tilde f$ does not depend on the mean of the target, a special feature of the Gaussian setting.

3.2. Numerical examples

The basic message of the present paper is that the process in the scaling limit of an MCMC algorithm contains useful information that can be utilized to achieve significant savings in high dimensions.

In both examples presented below the problem is to estimate the mean of the first coordinate $\rho_d(f)$ , where $f(\mathbf{x}^d)\,:\!=x^d_1$ . A run of T steps of a well-tuned RWM algorithm with kernel $P_d$ , defined at the beginning of Section 2, started in stationarity, produces an RWM sample $\{\mathbf{X}^d_n\}_{n=1,2,\ldots, T}$ and an estimate

\[ \hat{\rho}_d(f)\,:\!=\dfrac{1}{T}\sum_{n=1}^{T}f(\mathbf{X}^d_n) \]

of $\rho_d(f)$ . Take $\skew3\tilde{f}$ to be the associated solution of the Poisson equation, that we obtain numerically in the first example in Section 3.2.1 and using formula (3.1) in the second example in Section 3.2.2. In both cases we estimate the required unknown quantities ( $\rho_d(\kern1.8ptf)$ and $\Sigma$ ) from the sample $\{\mathbf{X}^d_n\}_{n=1,2,\ldots, T}$ .

Using the same sample, define

\[ \tilde{\rho}_d(\kern1.8ptf)\,:\!=\dfrac{1}{T}\sum_{n=1}^{T}f(\mathbf{X}^d_n). \]

Since the function $P_d\skew3\tilde{f}-\skew3\tilde{f}$ is not accessible in closed form, for every $n\leq T$ we use i.i.d. Monte Carlo to estimate the value $(P_d\skew3\tilde{f}-\skew3\tilde{f})(\mathbf{X}^d_n)$ as

\[ \dfrac{1}{n_{\mathrm{MC}}}\sum_{j=1}^{n_{\mathrm{MC}}} \bigg(1\wedge \dfrac{\rho_d(\mathbf{Y}^{d,j}_n}{\rho_d(\mathbf{X}^d_n)}\bigg) (\skew3\tilde{f}(\mathbf{Y}^{d,j}_n)-\skew3\tilde{f}(\mathbf{X}^d_n)),\]

where $\mathbf{Y}^{d,1}_n,\ldots,\mathbf{Y}^{d,n_{\mathrm{MC}}}_n$ is an i.i.d. sample of size $n_{\mathrm{MC}}$ from $N(\mathbf{X}^d_n,l^2/d\cdot I_d)$ . This estimation step can be parallelized (i.e. run on $n_{\mathrm{MC}}$ cores simultaneously).

We measure the variance reduction due to the postprocessing above by comparing the mean square errors of $\hat{\rho}_d(\kern1.8ptf)$ and $\tilde{\rho}_d(\kern1.8ptf)$ as estimators of $\rho_d(\kern1.8ptf)$ over $n_R$ independent runs of the RWM chain,

(3.2) \begin{equation} \text{VR}(\rho,f)\,:\!=\dfrac{\sum_{k=1}^{n_R}((\hat{\rho}_d(\kern1.8ptf))_k-\rho_d(\kern1.8ptf))^2}{\sum_{k=1}^{n_R}((\tilde{\rho}_d(\kern1.8ptf))_k-\rho_d(\kern1.8ptf))^2}, \end{equation}

where $ (\hat{\rho}_d(\kern1.8ptf))_k $ and $ (\tilde{\rho}_d(\kern1.8ptf))_k $ are the averages in the kth run of the chain. Heuristically, this means the estimator $\tilde{\rho}_d(\kern1.8ptf)$ of $\rho_d(\kern1.8ptf)$ based on T sample points is as good as estimator $\hat{\rho}_d(\kern1.8ptf)$ based on $\text{VR}(\rho,f)\cdot T$ sample points.

3.2.1. Multi-modal product target.

To verify that what theory predicts also happens in practice, we first present an example of an i.i.d. target with each coordinate a bimodal mixture of two Gaussian densities. Given the results in Table 1, we wish to highlight the robustness of the method with respect to numerically estimating $\skew3\tilde{f}$ and $(P_d\skew3\tilde{f}-\skew3\tilde{f})(\mathbf{X}^d_n)$ for each $n\leq T$ .

Table 1:

Variance reduction for different dimensions d and values of $n_{\mathrm{MC}}$ .

Let $\rho$ be a mixture of two normal densities $N(\mu_1,\sigma_1^2)$ and $N(\mu_2,\sigma_2^2)$ , with the first arising in the mixture with probability $2/5$ and $\mu_1=-3$ , $\mu_2=4$ , and $\sigma_1=\sigma_2=7/4$ . The potential of the density $\rho$ has two wells and is in a well-known class arising in models of molecular dynamics; see e.g. [Reference Duncan, Lelièvre and Pavliotis11, Section 5.4]. Note that the corresponding density $\rho_d$ on ${\mathbb{R}}^d$ , defined in the introduction, has $2^{d}$ modes.

With variance reduction (3.2) we measure how much the estimator $\tilde{\rho}_d(\kern1.8ptf)$ outperforms the estimator $\hat{\rho}_d(\kern1.8ptf)$ of the mean of the first coordinate $\rho_d(\kern1.8ptf)=6/5$ . The results across a range of dimensions d and values of the parameter $n_{\mathrm{MC}}$ (that corresponds to the accuracy of the estimation of $P_d\skew3\tilde{f}-\skew3\tilde{f}$ ) are presented in Table 1. All the entries were computed using $n_R=500$ independent runs of length $T=2\times 10^5$ .

To numerically solve the Poisson equation in (2.5), substitute the derivatives of f and $\log(\rho)$ with symmetric finite differences and use the estimate $\hat{\rho}(\kern1.8ptf)$ for $\rho(\kern1.8ptf)$ . Recall that the standard deviation of the proposal in our RWM algorithm is $l/\sqrt{d}$ . The solver uses a grid of hundred points equally spaced in the interval

\[ \Big[\min_{n\leq T }\mathbf{X}^d_{n,1}- 3 \cdot l/\sqrt{d}, \max_{n\leq T}\mathbf{X}_{n,1}+3\cdot l/\sqrt{d}\Big], \]

where $\mathbf{X}^d_{n,1}$ is the first coordinate of the nth sample point. We use the Moore–Penrose pseudo-inverse as the linear system is not of full rank. Finally, we take $\skew3\tilde{f}$ to be the linear interpolation of the solution on the grid. Note that this is a crude approximation of the solution of (2.5), which does not exploit analytical properties of either f or $\rho$ .

The results in Table 1 contain a lot of noise due to numerically solving the ODE, using an approximation $\hat{\rho}_d(\kern1.8ptf)$ for $\rho_d(\kern1.8ptf)$ and using i.i.d. Monte Carlo to estimate $\tilde{\rho}_d(\kern1.8ptf)$ . It is interesting to note that despite these additional sources of error, the variance reduction is considerable and behaves as the theoretical results predict. The estimator $\tilde{\rho}_d(\kern1.8ptf)$ improves with dimension, and with $n_{\mathrm{MC}}$ . Increasing $n_{\mathrm{MC}}$ , however, has a diminishing effect, which is particularly clear in the case $d=5$ . Due to the asymptotic nature of our result we can only expect limited gain for any fixed d, even if $P_d\skew3\tilde{f}-\skew3\tilde{f}$ could be evaluated exactly (corresponding to $n_{\mathrm{MC}}=\infty$ ).

3.2.2. Bimodal non-product target.

Can the theoretical findings of this paper help us construct control variates in more realistic cases with non-product target densities? It is unreasonable to expect a simple general answer to this question. A more realistic approach for future work seems to be trying to establish specific forms of control variates that work well for classes of targets of a certain type. We briefly explore one such instance in this section. Sections 3.1.4 and 3.1.5 as well as the results in Table 2 suggest that we can construct useful control variates when the target is close to Gaussian.

Table 2:

Variance reduction for different dimensions d and distances between modes h.

Let $\mu_{d,h}$ be a d-dimensional vector with entries $(h/2,0,\ldots,0)$ for $h\geq 0$ and let $\Sigma^{(d)}$ be a $d\times d$ covariance matrix with the largest eigenvalue equal to $\lambda=25$ , with the corresponding eigenvector being $ (1,1,\ldots, 1) $ and all other eigenvalues being equal to one. Take $\Pi_{d,h}$ to be the mixture of two d-dimensional normal densities $N({-}\mu_{d,h},\Sigma^{(d)})$ and $N(\mu_{d,h},\Sigma^{(d)})$ , both arising in the mixture with probability $1/2$ .

We wish to estimate the mean of the first coordinate $\Pi_{d,h}(\kern1.8ptf)=0$ (for $f(\mathbf{x}^d)=x^d_1$ ). To produce a control variate we simply pretend we are dealing with a Gaussian target instead of $\Pi_{d,h}$ . Let $\Sigma^{\Pi_{d,h}}$ be the covariance of $\Pi_{d,h}$ and let $\hat\Sigma^{\Pi_{d,h}}$ be an estimate of $\Sigma^{\Pi_{d,h}}$ obtained from the RWM sample $\{\textbf{X}^d_n\}_{n=1,2,\ldots,T}$ . Define $\skew3\tilde{f}_{d,h}$ as in (3.1) using $\hat\Sigma^{\Pi_{d,h}}$ . We compare the performance of estimators $\hat{\Pi}_{d,h}(\kern1.8ptf)$ and $\tilde{\Pi}_{d,h}(\kern1.8ptf)$ of $\Pi_{d,h}(\kern1.8ptf)=0$ , respectively defined as

\[ \dfrac{1}{T} \sum_{n=1}^Tf(\textbf{X}^d_n) \quad\text{and}\quad \dfrac{1}{T} \sum_{n=1}^T (\kern1.8ptf+dP_d\skew3\tilde{f}_{d,h}-d\skew3\tilde{f}_{d,h})(\textbf{X}^d_n), \]

according to variance reduction (3.2).

Table 2 shows the results across a range of dimensions d and distances between modes h, which measures the ’non-Gaussianity’ of the target. Note that when $h=0$ the target is Gaussian $N(0,\Sigma^{(d)})$ , which we include to demonstrate the validity of control variate (3.1) for Gaussian targets. All the entries were calculated using $n_R=500$ independent runs of length $T=2\times 10^5$ and $n_{\mathrm{MC}}=50$ i.i.d. Monte Carlo steps for computing $P_d\skew3\tilde{f}-\skew3\tilde{f}$ at each time step.

The quality of results decays with dimension because the proposal is scaled as $1/d$ in each coordinate. This results in the first coordinate mixing more slowly and for $h\neq 0$ also being less able to cross between modes, hence our estimate $\hat\Sigma^{\Pi_{d,h}}$ of the covariance becomes worse as we are working with fixed RWM sample length T. When $h=10$ (and some cases of $h=8$ ), it is unlikely that the RWM sample will reach the other mode at all, which results in no gain from the method.

If we use the true covariance $\Sigma^{\Pi_{d,h}}$ of the target in the control variate (3.1), instead of learning it from the sample $\hat\Sigma^{\Pi_{d,h}}$ , the corresponding results for $d=50$ are presented in Table 3.

Table 3:

Variance reduction for dimension $d=50$ and different distances between modes h using the true covariance of the target.

Unsurprisingly the estimator $\tilde{\Pi}_{d,h}(\kern1.8ptf)$ does not perform well when the distance between modes h is large. Interestingly, though, the method does offer considerable gain in cases $h\,{=}\,2$ and $h=4$ , and even a noticeable gain in $h=6$ . For $h=4$ and $h=6$ the target is already clearly bimodal and different from the Gaussian; the RWM sample stays in the same mode for hundreds, respectively thousands of time steps at a time.

4.1. Outline of the proof of Theorem 2.2

We start by specifying sets $\mathcal{A}_d\subset {\mathbb{R}}^d$ that have large probability under $\rho_d$ . We need the following fact.

Proposition 4.1. There exists a constant $c_A>0$ such that the following open subset of ${\mathbb{R}}$ satisfies $\rho(A)>0$ :

\[ A\,:\!=\{x\in{\mathbb{R}}\colon |\log(\rho)''(x)|< (\log(\rho)'(x))^2,1/c_A<|\log(\rho)'(x)|< c_A\}. \]

Let A satisfy the conclusion of Proposition 4.1 and recall the notation $\rho(\kern1.8ptf)=\int_{{\mathbb{R}}}f(x)\rho(x) \,{\mathrm{d}} x$ for any appropriate function $f \colon {\mathbb{R}}\to{\mathbb{R}}$ . Recall $J=\rho(((\log\rho)')^2)=-\rho((\log \rho)'')$ , where the equality follows from assumptions $\log(\rho)\in\mathcal{S}^4$ and (2.3). Define the sets $\mathcal{A}_d$ as follows.

Definition 4.1. Any $\mathbf{x}^d\in{\mathbb{R}}^d$ is in $\mathcal{A}_d$ if and only if the following four assumptions hold:

(4.1) \begin{align} \dfrac{1}{d-1}\sum_{i=2}^d\,{\mathrm{e}}^{|x^d_i|} & < 2\int_{\mathbb{R}} \,{\mathrm{e}}^{|x|}\rho(x) \,{\mathrm{d}} x,\end{align}

(4.2) \begin{align}* \dfrac{1}{d-1}\sum_{i=2}^d1_{A}(x^d_i) & > \dfrac{\rho(A)}{2},\end{align}

(4.3) \begin{align} \dfrac{1}{d-1}\bigg|\sum_{i=2}^d (\log(\rho)'(x^d_i))^2-J\bigg| & <\dfrac{a_d}{\sqrt{d}} \sqrt{3\int_{{\mathbb{R}}} ( (\log(\rho)'(x))^2-J)^2\rho(x) \,{\mathrm{d}} x},\end{align}

(4.4) \begin{align}* \dfrac{1}{d-1}\bigg|\sum_{i=2}^d \log(\rho)''(x^d_i)+J\bigg| & <\dfrac{a_d}{\sqrt{d}} \sqrt{3\int_{{\mathbb{R}}} (\log(\rho)''(x)+J)^2\rho(x) \,{\mathrm{d}} x}. \end{align}

Using the theory of large deviations and classical inequalities, the proof of the proposition bounds the probabilities of sets, where each of the above four assumptions in Definition 4.1 fails (see Sections 4.2 and 5.2 below for details).

Pick any $f\in\mathcal{S}^3$ and express the generator $\mathcal{G}_d$ , defined in (2.1), as follows:

\[\mathcal{G}_d\kern1.5ptf(\mathbf{x}^d)=d\cdot \mathbb{E}_{Y^d_1}\bigg[ (\kern1.8ptf(\mathbf{Y}^d)-f(\mathbf{x}^d)) \mathbb{E}_{\mathbf{Y}^d-}\bigg[1\wedge\dfrac{\rho_d(\mathbf{Y}^d)}{\rho_d(\mathbf{x}^d)}\bigg]\bigg], \quad\mathbf{x}^d\in{\mathbb{R}}^d, \]

where $\mathbb{E}_{\mathbf{Y}^d-} [\cdot]$ is the expectation with respect to all the coordinates of the proposal $\mathbf{Y}^d$ in ${\mathbb{R}}^d$ , except the first one (identify $f\in L^1(\rho)$ with $f\in L^1(\rho_d)$ by ignoring the last $d-1$ coordinates). The strategy of the proof of Theorem 2.2 is to define a sequence of operators, ‘connecting’ $\mathcal{G}_d$ and $\mathcal{G}$ , such that each approximation can be controlled for $f\in\mathcal{S}^3$ and $\mathbf{x}^d\in\mathcal{A}_d$ .

First, for $f\in\mathcal{S}^3$ , define

(4.5) \begin{equation} \tilde{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)\,:\!=d\cdot \mathbb{E}_{Y^d_1} [ (\kern1.8ptf(Y^d_1)-f(x^d_1))\beta(\mathbf{x}^d,Y^d_1)],\quad \mathbf{x}^d\in{\mathbb{R}}^d, \end{equation}

where, for any $y\in{\mathbb{R}}$ ,

(4.6) \begin{equation} \beta(\mathbf{x}^d, y)\,:\!=\mathbb{E}_{\mathbf{Y}^d-} \bigg[1\wedge \exp\bigg(\log(\rho)'(x^d_1)( y-x^d_1)+\sum_{i=2}^d K(x^d_i, Y^d_i)\bigg)\bigg],\quad\mathbf{x}^d\in{\mathbb{R}}^d, \end{equation}

and for any $ (x,y)\in{\mathbb{R}}^2 $ we define

(4.7) \begin{equation} K(x, y)\,:\!=\log(\rho)'(x)( y-x)+\dfrac{\log(\rho)''(x)}{2}( y-x)^2+\dfrac{ (\log(\rho)'(x))^31_{A}(x)}{3}( y-x)^3. \end{equation}

In (4.7), the set A satisfies the conclusion of Proposition 4.1 and the coefficient before $ ( y\,{-}\,x)^3 $ is chosen so that it is uniformly bounded for all $x\in {\mathbb{R}}$ . This property plays an important role in proving that we have uniform control over the supremum norms of certain densities; see Lemmas 4.1 and 4.2 below. We can now prove the following.

Proposition 4.3. There exists a constant C such that, for every $f\in\mathcal{S}^3$ and all $d\in{\mathbb{N}}$ , we have

\[ |\mathcal{G}_d\kern1.5ptf(\mathbf{x}^d)-\tilde{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)|\leq C \|\:f'\|_{\infty,1/2} \,{\mathrm{e}}^{|x^d_1|}d^{-1/2} \quad \text{{for all} }\mathbf{x}^d\in\mathcal{A}_d. \]

The proof of Proposition 4.3 relies only on the elementary bounds from Section 5.1 below. The idea is to use the Taylor series of $\log(\rho)(Y_i)$ around $x^d_i$ for every $i\in\{1,\ldots,d\}$ and then prove that modifying terms of order higher than two if $i\in\{2,\ldots,d\}$ (resp. one if $i=1$ ) is inconsequential.

Define the operator $\hat{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)$ for any $f\in\mathcal{S}^3$ and $\mathbf{x}^d\in{\mathbb{R}}^d$ by

(4.8) \begin{align} \hat{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)&\,:\!= \dfrac{l^2}{2}f''(x^d_1)\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\sum_{i=2}^d K(x^d_i, Y^d_i)}\Big] \notag \\ &\quad\, +l^2f'(x^d_1)\log(\rho)'(x^d_1)\mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\sum_{i=2}^d K(x^d_i, Y^d_i)}1_{\{\sum_{i=2}^d K(x^d_i, Y^d_i)<0\}}\Big]. \end{align}

We can now prove the following fact.

Proposition 4.4. There exists a constant C such that, for every $f\in\mathcal{S}^3$ , and all $d\in{\mathbb{N}}$ , we have

\[ |\hat{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)-\tilde{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)|\leq C\bigg(\sum_{i=1}^3\|\:f^{(i)}\|_{\infty,1/2}\bigg)\,{\mathrm{e}}^{|x^d_1|}d^{-1/2}\quad \text{{for all} }\mathbf{x}^d\in\mathcal{A}_d. \]

Note that if we freeze the coordinates $x_2^d,\ldots,x_d^d$ in $\mathbf{x}^d$ , the operator mapping $f\in\mathcal{S}^3$ to $x_1^d\mapsto \hat{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)$ generates a one-dimensional diffusion with coefficients of the same functional form as in $\mathcal{G}$ , but with slightly modified parameter values. The proof of Proposition 4.4 is based on the third and second degree Taylor’s expansion of $y\mapsto f( y)$ and $y\mapsto \beta(\mathbf{x}^d,y)$ (around $x^d_1$ ), respectively, applied to the definition of $\tilde{\mathcal{G}}_d$ in (4.5). The difficult part in proving that the remainder terms can be omitted consists of controlling $ {\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,y) $ , as this entails bounding the supremum norm of the density of $\sum_{i=2}^d K(x^d_i, Y^d_i)$ uniformly in d. Condition (4.2), which forces a portion of the coordinates $x^d_i$ of $\mathbf{x}^d$ to be in the set A where the densities of the corresponding summands $K(x^d_i,Y^d_i)$ can be controlled, was introduced for this purpose. The details, explained in Sections 4.2 and 5.3 below, rely crucially on the optimal version of Young’s inequality.

Introduce the following normal random variable with mean

\[ \mu_{\mathcal{N}}(\mathbf{x}^d)=\dfrac{l^2}{2d}\sum_{i=2}^d\log(\rho)''(x^d_i) \]

and variance

\[ \sigma^2_{\mathcal{N}}(\mathbf{x}^d)=\dfrac{l^2}{d}\sum_{i=2}^d (\log(\rho)'(x^d_i))^2{,} \]

that is,

(4.9) \begin{equation} \mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)\,:\!=\dfrac{l^2}{2d}\sum_{i=2}^d\log(\rho)''(x^d_i)+\sum_{i=2}^d\log(\rho)'(x^d_i)(Y^d_i-x^d_i). \end{equation}

Define the operator $\breve{\mathcal{G}}_d\kern1.5ptf(\mathbf{x}^d)$ for $f\in\mathcal{S}^3$ and $\mathbf{x}^d\in{\mathbb{R}}^d$ by

\begin{align*} \breve{\mathcal{G}}_df(\mathbf{x}^d)&\,:\!= \frac{l^2}{2}f''(x^d_1)\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big] \\ & \quad\, + l^2f'(x^d_1)\log(\rho)'(x^d_1)\mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big]. \end{align*}

Proposition 4.5. There exists a constant C such that, for every $f\in\mathcal{S}^3$ and all $d\in{\mathbb{N}}$ , we have

\[ |\breve{\mathcal{G}}_df(\mathbf{x}^d)-\hat{\mathcal{G}}_df(\mathbf{x}^d)|\leq C\bigg(\sum_{i=1}^2\|\:f^{(i)}\|_{\infty,1/2}\bigg) \,{\mathrm{e}}^{|x^d_1|}d^{-1/2} \quad\text{{for all} }\mathbf{x}^d\in\mathcal{A}_d. \]

First we show that

\[ \Big|\mathbb{E}_{\mathbf{Y}^d-}\Big[1\wedge {\mathrm{e}}^{\sum_{i=2}^d K(x^d_i, Y^d_i)}\Big]-\mathbb{E}_{\mathbf{Y}^d-}\Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big] \Big| \]

is small (Lemma 4.4 below). Proving that

\[ \mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\sum_{i=2}^d K(x^d_i, Y^d_i)}1_{\{\sum_{i=2}^d K(x^d_i, Y^d_i)<0\}}\Big]\quad\text{and}\quad \mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big] \]

are close is challenging, as it requires showing that the supremum norm of the difference between the distributions of $\mathcal{N}(\mathbf{x}^{d},\mathbf{Y}^d)$ and $\sum_{i=2}^{d} K(x^d_i, Y^d_i)$ decays as $d^{-1/2}$ uniformly in its argument. The proof of this fact mimics the proof of the Berry–Esseen theorem and relies on the closeness of the CFs (characteristic functions) of $\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)$ and $\sum_{i=2}^d K(x^d_i, Y^d_i)$ . The particular form of K(x,Y) makes it possible to explicitly calculate the CF of K(x,Y), if $x\notin A$ , and bound it appropriately, if $x\in A$ . The details are explained in Sections 4.2 and 5.4 below.

Since $\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)$ is normal, it is possible to explicitly calculate the expectations

\[ \mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big] \quad\text{and}\quad \mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big] \]

(see [Reference Roberts, Gelman and Gilks30, Proposition 2.4]). Using these formulae, Proposition 4.6, which implies Theorem 2.2, can be deduced from assumptions (4.3)–(4.4).

Proposition 4.6. There exists a constant C such that, for every $f\in\mathcal{S}^3$ and all $d\in{\mathbb{N}}$ , we have

\[ |\mathcal{G}f(x^d_1)-\breve{\mathcal{G}}_df(\mathbf{x}^d)|\leq C\bigg(\sum_{i=1}^2\|\:f^{(i)}\|_{\infty,1/2}\bigg) \,{\mathrm{e}}^{|x^d_1|}\dfrac{a_d}{\sqrt{d}} \quad\text{{for all} }\mathbf{x}^d\in\mathcal{A}_d.\]

4.2. Proof of Theorem 2.2

Proof of Proposition 4.1. Let

\[ \skew2\tilde{A}\,:\!=\{ x\in{\mathbb{R}}\colon |\log(\rho)''(x)|< (\log(\rho)'(x))^2\}. \]

It suffices to show that the open set $\tilde A$ is not empty, since

\[ \skew2\tilde{A}=\cup_{n\in{\mathbb{N}}}\Big(\skew2\tilde{A}\cap\Big\{x\in{\mathbb{R}}\colon \dfrac{1}{n}{<} |\log(\rho)'(x)|{<}n\Big\}\Big), \]

so for some large $n_0$ the open set

\[ \skew2\tilde{A}\cap\Big\{x\in{\mathbb{R}}\colon \dfrac{1}{n_0}{<} |\log(\rho)'(x)|{<} n_0\Big\} \]

must have positive Lebesgue measure and we can take $c_A\,:\!=n_0$ .

Assume that $\skew2\tilde{A}=\emptyset$ , i.e. $|u'|\geq u^2$ on ${\mathbb{R}}$ , where $u\,:\!=\log(\rho)'$ . Since $\rho$ satisfies (2.3), there exists $x_0<0$ and $C>0$ such that $u>C$ on the interval $({-}\infty,x_0)$ . Moreover, since $|u'|\geq u^2>C^2>0$ , u’ has no zeros on $({-}\infty,x_0)$ and satisfies either $u'\geq u^2$ or $-u'\geq u^2$ on the half-infinite interval. Since $ (1/u)'=-u'/u^2 $ , integrating the inequalities $-u'/u^2\leq -1$ or $-u'/u^2\geq1$ from any $x\in({-}\infty,x_0)$ to $x_0$ , we get $1/u(x_0)+x_0-x\leq 1/u(x)$ and $1/u(x_0)+x-x_0\geq 1/u(x)$ . Since by assumption $0<1/u<1/C$ on $ ({-}\infty,x_0) $ , we get a contradiction in both cases.

Proof of Proposition 4.2. Let $B^d_1$ , $B^d_2$ , $B^d_3$ , and $B^d_4$ be the subsets of ${\mathbb{R}}^d$ where assumptions (4.1), (4.2), (4.3), and (4.4) are not satisfied, respectively. Note that

\[ {\mathbb{R}}^d\setminus \mathcal{A}_d=B^d_1\cup B^d_2\cup B^d_3 \cup B^d_4. \]

Recall that by (2.3), L’Hôpital’s rule implies

\[ \lim_{|x|\to\infty}\dfrac{\log\rho(x)}{x}\to-\infty \]

and hence $\rho({\mathrm{e}}^{s|x|})<\infty$ for any $s>0$ . Since $\{a_d\}_{d\in{\mathbb{N}}}$ is sluggish, there exists $n\in{\mathbb{N}}$ such that $a_d\leq \sqrt{n\log d}$ for all $d\in{\mathbb{N}}$ . Then, by Proposition 5.4 applied to functions $x\mapsto ({\mathrm{e}}^{|x|}-\rho({\mathrm{e}}^{|x|}))/\rho({\mathrm{e}}^{|x|})$ and $x\mapsto 2(\rho(A)-1_A(x))/\rho(A)$ , respectively, there exist constants $c_1', c_2'$ such that the inequalities $\rho(B^d_1)\leq c_1'd^{-n}\leq c_1'\,{\mathrm{e}}^{-a_d^2}$ and $\rho(B^d_2)\leq c_2'd^{-n}\leq c_2'\,{\mathrm{e}}^{-a_d^2}$ hold for all $d\in{\mathbb{N}}$ .

Likewise, there exist constants $c_3', c_4'$ such that $\rho(B^d_3)\leq c_3' \,{\mathrm{e}}^{-a^2_d}$ and $\rho(B^d_4)\leq c_4' \,{\mathrm{e}}^{-a^2_d}$ . This follows by Proposition 5.3, applied to the sequence $\{a_d\}_{d\in{\mathbb{N}}}$ and functions

\[ g_3(x)\,:\!= (\log(\rho)'(x))^2-J \]

(with $t\,:\!=\sqrt{3\rho(g_3^2)}$ ) and

\[ g_4(x)\,:\!=\log(\rho)''(x)+J \]

(with $t\,:\!=\sqrt{3\rho(g_4^2)}$ ), respectively. Hence

\[ \rho({\mathbb{R}}^d\setminus \mathcal{A}_d)\leq \rho(B^d_1)+\rho(B^d_2)+\rho(B^d_3 )+\rho(B^d_4)\leq c_1 \,{\mathrm{e}}^{-a_d^2} \]

for $c_1\,:\!=\max\{c_1',c_2',c_3',c_4'\}$ .

Proof of Proposition 4.3. Pick an arbitrary $\mathbf{x}^d\in \mathcal{A}_d$ and recall that $\alpha(\mathbf{x}^d,\mathbf{Y}^d)$ is defined in (2.1). Since $|1\wedge {\mathrm{e}}^x-1\wedge {\mathrm{e}}^y|\leq |x-y|$ for all $x,y\in{\mathbb{R}}$ , for every realization $Y^d_1$ , Taylor’s theorem implies

(4.10) \begin{equation} |\mathbb{E}_{\mathbf{Y}^d-}[\alpha(\mathbf{x}^d,\mathbf{Y}^d)]-\beta(\mathbf{x}^d, Y^d_1)|\leq |\log(\rho)''(W^d_1)|(Y^d_1-x^d_1)^2+T^d_1(\mathbf{x}^d)+T^d_2(\mathbf{x}^d), \end{equation}

where $W^d_1$ satisfies

\[ \log(\rho)''(W^d_1)(Y^d_1-x^d_1)^2/2=\log(\rho)(Y^d_1)-\log(\rho)(x^d_1)-\log{\rho}'(x^d_1)(Y^d_1-x^d_1) \]

and

\begin{align*} T^d_1(\mathbf{x}^d)&\,:\!= \dfrac{1}{6}\mathbb{E}_{\mathbf{Y}^d-}\bigg[\bigg|\sum_{i=2}^d (\log(\rho)'''(x^d_i)-2(\log(\rho)'(x^d_i))^31_{A}(x^d_i))(Y_i-x^d_i)^3\bigg|\bigg], \\ T^d_2(\mathbf{x}^d)&\,:\!= \dfrac{1}{24}\mathbb{E}_{\mathbf{Y}^d-}\bigg[\sum_{i=2}^d|\log(\rho)^{(4)}(Z^d_i)|(Y_i-x^d_i)^4\bigg]. \end{align*}

Here $Z^d_i$ satisfies

\[ \dfrac{1}{4!}\log(\rho)^{(4)}(Z^d_i)(Y^d_i-x^d_i)^4=\log(\rho)(Y^d_i)-\sum_{j=0}^{3}\dfrac{1}{j!}\log(\rho)^{(j)}(x^d_i)(Y^d_i-x^d_i)^j\]

for any $2\leq i\leq d$ . Recall that $Y^d_i-x^d_i$ is normal $N(0,l^2/d)$ , for some constant $l>0$ , and $\log(\rho)\in\mathcal{S}^4$ . Hence we may apply Proposition 5.2 to the function

\[ x\mapsto \log(\rho)'''(x)-2(\log(\rho)'(x))^31_{A}(x) \]

to get

\[ T^d_1(\mathbf{x}^d)\leq C_1 \bigg(\dfrac{l^6}{d^3} \sum_{i=2}^d \,{\mathrm{e}}^{|x^d_i|}\bigg)^{1/2} \]

for some constant $C_1>0$ , independent of $\mathbf{x}^d$ . Since $\mathbf{x}^d\in \mathcal{A}_d$ , the assumption in (4.1) yields $T^d_1(\mathbf{x}^d)\leq C_1 l^3 (2\rho({\mathrm{e}}^{|x|}))^{1/2}/d$ . Similarly, we apply Proposition 5.1 (with $f=\log\rho$ , $n=k=4$ , $m=1$ , $s=1$ and $\sigma^2=l^2/d$ ) and assumption (4.1) to get

\[ T^d_2(\mathbf{x}^d)\leq C_2 d^{-2}\sum_{i=2}^d\,{\mathrm{e}}^{|x^d_i|}\leq C_2 d^{-1} \]

for some constant $C_2>0$ and all $\mathbf{x}^d\in \mathcal{A}_d$ .

Recall $f\in\mathcal{S}^3$ and let $\skew2\tilde{W}^d_1$ be as in Proposition 5.1, satisfying

\[ f'(\skew2\tilde{W}^d_1)(Y^d_1-x^d_1)=f(Y^d_1)-f(x^d_1). \]

Let $C>0$ be such that $T^d_1(\mathbf{x}^d)+T^d_2(\mathbf{x}^d)\leq Cd^{-1}$ for all $\mathbf{x}^d\in \mathcal{A}_d$ . The bound in (4.10), Taylor’s theorem applied to f, and Cauchy’s inequality yield

\begin{align*} |\mathcal{G}_df(\mathbf{x}^d)-\tilde{\mathcal{G}}_df(\mathbf{x}^d)| & \leq d \mathbb{E}_{Y^d_1} [ | f(Y^d_1)-f(x^d_1)| (|\log(\rho)''(W^d_1)|(Y^d_1-x^d_1)^2+Cd^{-1})] \\ & = d \mathbb{E}_{Y^d_1} [ | f'(\skew2\tilde{W}^d_1)\log(\rho)''(W^d_1)(Y^d_1-x^d_1)^3|]+C\mathbb{E}_{Y^d_1} [ | f'(\skew2\tilde{W}^d_1)(Y^d_1-x^d_1)|] \\ & \leq d (\mathbb{E}_{Y^d_1} [ | f'(\skew2\tilde{W}^d_1)^2(Y^d_1-x^d_1)^3|]\mathbb{E}_{Y^d_1} [ | (\log(\rho)'')^2(W^d_1)(Y^d_1-x^d_1)^3|])^{1/2} \\ & \quad\, +C\mathbb{E}_{Y^d_1} [ | f'(\skew2\tilde{W}^d_1)(Y^d_1-x^d_1)|]\\ &\leq \skew3\bar C d(\|\:f'\|_{\infty,1/2}^2 \,{\mathrm{e}}^{|x_1^d|} d^{-3/2} \cdot {\mathrm{e}}^{|x_1^d|}d^{-3/2})^{1/2} + \skew3\bar C \|\:f'\|_{\infty,1/2}\,{\mathrm{e}}^{|x_1^d|} d^{-1/2}. \end{align*}

The last inequality follows by three applications of Proposition 5.1, where $\skew3\bar C>0$ is a constant that does not depend on f or $\mathbf{x}^d\in \mathcal{A}_d$ . This concludes the proof of the proposition.

Before tackling the proof of Proposition 4.4, we need the following three lemmas. Recall that K(x,Y) is defined in (4.7) and the set A satisfies the conclusion of Proposition 4.1.

Proof. Existence of $q_x$ follows from (4.7) and Proposition 5.6. By Proposition 4.1 we have

\[ |\log(\rho)''(x)|< (\log(\rho)'(x))^2\quad\text{and}\quad c_A> |\log(\rho)'(x)|>1/c_A. \]

Consider the polynomial

\[ y\mapsto p( y)\,:\!=\log(\rho)'(x)\cdot y+\log(\rho)''(x)\cdot y^2/2+ (\log(\rho)'(x))^3\cdot y ^3/3. \]

By (4.7), $p(Y-x)=K(x,Y)$ . Since

\[ p'( y)=\log(\rho)''(x)\cdot y+\log(\rho)'(x)(1+\log(\rho)'(x)^2\cdot y^2), \]

we have

\begin{align*} |p'( y)| &\geq |\log(\rho)'(x)|(1+\log(\rho)'(x)^2\cdot y^2) - |\log(\rho)''(x)|\,|y| \\ &> |\log(\rho)'(x)|(1-|\log(\rho)'(x)\cdot y|+|\log(\rho)'(x) \cdot y|^2)\\ &> \dfrac{3}{4c_A}, \end{align*}

where the second inequality holds since $ |\log(\rho)''(x)|< (\log(\rho)'(x))^2$ and the third follows from $\inf_{z\in{\mathbb{R}}}\{1-|z|+z^2\}=3/4$ and $ |\log(\rho)'(x)|>1/c_A$ . The lemma now follows by Proposition 5.7.

Recall that the proposal is normal $\mathbf{Y}^d=(Y_1^d,\ldots,Y_d^d)\sim N(\mathbf{x}^d,l^2/d\cdot I_d)$ .

Proof. Fix $\mathbf{x}^d\in\mathcal{A}_d$ and, for each i, let $q_i$ denote the density of $K(x^d_i,Y^d_i)$ as in the previous lemma. Since the components of $\mathbf{Y}^d$ are i.i.d., we have $\mathbf{q}^d_{\mathbf{x}^d}=\mathop{\mbox{\ast}}_{i=2}^d q_i=q_A \mathop{\mbox{\ast}} q_{{\mathbb{R}}\setminus A}$ , where $q_A\,:\!= \mathop{\mbox{\ast}}_{x^d_i\in A}q_i$ and $q_{{\mathbb{R}}\setminus A}\,:\!=\mathop{\mbox{\ast}}_{x^d_i\notin A}q_i$ . By the definition of convolution and the fact that $q_{{\mathbb{R}}\setminus A}$ is a density, it follows that $\|\mathbf{q}^d_{\mathbf{x}^d}\|_\infty\leq \|q_A\|_\infty\|q_{{\mathbb{R}}\setminus A}\|_1=\|q_A\|_\infty$ . By Lemma 4.1 there exists $C>0$ such that, for any $d\in{\mathbb{N}}$ , we have $\|q_i\|_\infty<C\sqrt{d}$ if $x^d_i\in A$ . Condition (4.2) implies there are at least $ (d-1)\rho(A)/2 $ factors in the convolution $q_A=\mathop{\mbox{\ast}}_{x^d_i\in A}q_i$ . Hence Proposition 5.5 applied to $q_A$ yields

\[ \|\mathbf{q}^d_{\mathbf{x}^d}\|_\infty\leq\|q_A\|_\infty\leq c \dfrac{C\sqrt{d}}{\sqrt{(d-1)\rho(A)/2}}. \]

This concludes the proof of the lemma.

Proof. Points (i) and (ii) follow from the definition in (4.6). Since $x\mapsto 1\wedge {\mathrm{e}}^x$ is Lipschitz (with Lipschitz constant 1) on ${\mathbb{R}}$ , the family of functions

\[ \bigg\{x\mapsto \dfrac{1}{h}(1\wedge \mathrm{e}^{x+h}- 1\wedge \mathrm{e}^x)\colon h\in\mathbb{R}\setminus\{0\}\bigg\} \]

is bounded by one and converges pointwise to $1_{\{x<0\}}\,{\mathrm{e}}^x$ for all $x\in{\mathbb{R}}\setminus\{0\}$ , as $h\to0$ . Hence the DCT implies that ${\partial}/{\partial y}\beta(\mathbf{x}^d,y)$ exists and can be expressed as

(4.11) \begin{equation} \log(\rho)'(x^d_1)\mathbb{E}_{\mathbf{Y}^d-}\Big[{\mathrm{e}}^{\log(\rho)'(x^d_1)( y-x^d_1) +\sum_{i=2}^dK(x^d_i,Y^d_i)}1_{\{\log(\rho)'(x^d_1)( y-x^d_1)+\sum_{i=2}^dK(x^d_i,Y^d_i)<0\}}\Big], \end{equation}

implying (iii) and (iv). Let $\Phi^d_{K}$ denote the distribution of $\sum_{i=2}^dK(x^d_i,Y^d_i)$ , and recall that by definition we have ${\mathrm{e}}^x 1_{\{x<0\}} = 1\wedge {\mathrm{e}}^x - 1_{\{x\geq 0\}}$ for all $x\in{\mathbb{R}}$ . Hence, by (4.11), it follows that

(4.12) \begin{equation} \dfrac{\partial}{\partial y}\beta(\mathbf{x}^d,y) = \log(\rho)'(x^d_1) (\beta(\mathbf{x}^d,y)-1+\Phi^d_{K} ({-}\log(\rho)'(x^d_1)( y-x^d_1))), \end{equation}

By Lemma 4.2, $\Phi_K^d$ is differentiable. Hence, by (4.12), ${\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,y)$ also exists and takes the form

\[ (\log(\rho)'(x^d_1))^2 (\beta(\mathbf{x}^d,y)-1 +\Phi^d_{K} ({-}\log(\rho)'(x^d_1)( y-x^d_1))-\mathbf{q}^d_{\mathbf{x}^d} ({-}\log(\rho)'(x^d_1)( y-x^d_1))).\]

Part (v) follows from this representation of ${\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,y)$ and Lemma 4.2.

Proof of Proposition 4.4. Fix an arbitrary $\mathbf{x}^d\in\mathcal{A}_d$ . Let $Z_1,W_1$ be random variables, as in Proposition 5.1, that satisfy

\begin{align*} f(Y^d_1)-f(x^d_1) & = f'(x^d_1)(Y^d_1-x^d_1)+\frac{f''(x^d_1)}{2}(Y^d_1-x^d_1)^2+\frac{f'''(Z_1)}{6}(Y^d_1-x^d_1)^3, \\ \beta(\mathbf{x}^d,Y^d_1) & = \beta(\mathbf{x}^d,x^d_1)+\frac{\partial}{\partial y}\beta(\mathbf{x}^d,x^d_1)(Y^d_1-x^d_1)+\frac{{\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,W_1)}{2}(Y^d_1-x^d_1)^2. \end{align*}

Then, by the definition of $\tilde{\mathcal{G}}_df(\mathbf{x}^d)$ in (4.5) and the fact that $Y_1^d-x_1^d\sim N(0,l^2/d)$ , we obtain

\begin{align*} \tilde{\mathcal{G}}_df(\mathbf{x}^d)&= \frac{l^2f''(x^d_1)}{2}\beta(\mathbf{x}^d,x^d_1)+l^2f'(x^d_j)\frac{\partial}{\partial y}\beta(\mathbf{x}^d,x^d_1) \\ &\quad\, + d \mathbb{E}_{Y^d_1} \bigg[ (\beta(\mathbf{x}^d,x^d_1)\frac{f'''(Z_1)}{6}+f'(x^d_1)\frac{{\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,W_1)}{2})(Y^d_1-x^d_1)^3\bigg] \\ &\quad\, + d \mathbb{E}_{Y^d_1}\bigg[\bigg(\frac{f''(x^d_1)}{2}\frac{{\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,W_1)}{2}+\frac{f'''(Z_1)}{6}\frac{\partial}{\partial y}\beta(\mathbf{x}^d,x^d_1)\bigg)(Y^d_1-x^d_1)^4\bigg] \\ &\quad\, + d \mathbb{E}_{Y^d_1}\bigg[\frac{f'''(Z_1)}{6}\frac{{\partial^2}/{\partial y^2}\beta(\mathbf{x}^d,W_1)}{2}(Y^d_1-x^d_1)^5\bigg]. \end{align*}

By parts (ii) and (iv) in Lemma 4.3 and the definition of $\hat{\mathcal{G}}_df(\mathbf{x}^d)$ in (4.8), we have

\[ \hat{\mathcal{G}}_df(\mathbf{x}^d)= \dfrac{l^2f''(x^d_1)}{2}\beta(\mathbf{x}^d,x^d_1)+l^2f'(x^d_j)\dfrac{\partial}{\partial y}\beta(\mathbf{x}^d,x^d_1). \]

The three expectations in the display above can each be bounded by a constant times

\[ \bigg(\sum_{i=1}^3\|\:f^{(i)}\|_{\infty,1/2}\bigg)\,{\mathrm{e}}^{|x^d_1|}d^{-1/2} \]

using Proposition 5.1 and Lemma 4.3. For instance, the first expectation can be bounded above using (v) in Lemma 4.3:

\[\dfrac{d}{6}\mathbb{E}_{Y^d_1} [| f'''(Z_1)|\,|Y^d_1-x^d_1|^3]+\dfrac{d (C_K+1)}{2}| f'(x^d_1)|\,|\log(\rho'(x^d_1))|^2\mathbb{E}_{Y^d_1} [|Y^d_1-x^d_1|^3].\]

Proposition 5.1 yields

\[ \dfrac{d }{6}\mathbb{E}_{Y^d_1} [| f'''(Z_1)|\,|Y^d_1-x^d_1|^3] \leq C_0 \,{\mathrm{e}}^{|x^d_1|} \|\:f'''\|_{\infty,1} d^{-1/2}\leq C_0 \,{\mathrm{e}}^{|x^d_1|} \|\:f'''\|_{\infty,1/2} d^{-1/2} \]

for some $C_0>0$ . Moreover,

\[ | f'(x^d_1)|\,|\log(\rho'(x^d_1))|^2\leq \|\:f'\|_{\infty,1/2}\|(\log(\rho)')^2\|_{\infty,1/2}\,{\mathrm{e}}^{|x^d_1|} \]

as $\log(\rho)\in\mathcal{S}^4$ and $f\in\mathcal{S}^3$ . Hence

\[ \dfrac{d (C_K+1)}{2}| f'(x^d_1)|\,|\log(\rho'(x^d_1))|^2\mathbb{E}_{Y^d_1} [|Y^d_1-x^d_1|^3] \leq C_1 \,{\mathrm{e}}^{|x^d_1|} \|\:f'\|_{\infty,1/2} d^{-1/2} \]

for some $C_1>0$ . Similarly, it follows that the second and third expectations above decay as $d^{-1}$ and $d^{-3/2}$ , respectively. This concludes the proof of the proposition.

Lemma 4.4. Recall that $\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)$ and $\sum_{i=2}^dK(x^d_i,Y^d_i)$ are defined in (4.9) and (4.7), respectively. Then there exists a constant C such that, for all $d\in{\mathbb{N}}$ , we have

\[\mathbb{E}_{\mathbf{Y}^d-}\bigg[\bigg|\sum_{i=2}^dK(x^d_i,Y^d_i)-\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)\bigg|\bigg]\leq Cd^{-1/2}\quad \text{{for all} }\mathbf{x}^d\in\mathcal{A}_d.\]

Proof. The difference in question is smaller than the sum of the following two terms:

\begin{align*} T^d_3(\mathbf{x}^d)&= \mathbb{E}_{\mathbf{Y}^d-}\bigg[\bigg|\sum_{i=2}^d\dfrac{\log(\rho)''(x^d_i)}{2}\bigg((Y^d_i-x^d_i)^2-\dfrac{l^2}{d}\bigg)\bigg|\bigg], \\ T^d_4(\mathbf{x}^d)&= \mathbb{E}_{\mathbf{Y}^d-}\bigg[\bigg|\sum_{i=2}^d\dfrac{ (\log(\rho)'(x^d_i))^31_{A}(x^d_i)}{3}(Y^d_i-x^d_i)^3\bigg|\bigg]. \end{align*}

Note that $X_i\,:\!=(Y^d_i-x^d_i)^2-l^2/d$ , $2\leq i\leq d$ , are zero mean i.i.d. with $\mathbb{E}[X_i^4] = 2l^4/d^2$ . Hence, as $\log(\rho)\in\mathcal{S}^4$ , we may apply Proposition 5.2 with the function $x\mapsto \log(\rho)''(x)$ and $X_2$ , for $2\leq i\leq d$ , to get

\[ T^d_3(\mathbf{x}^d)\leq \|\log(\rho)''\|_{\infty,1/2} \bigg(2(l^4/d^2)\sum_{i=2}^d \,{\mathrm{e}}^{|x^d_i|}\bigg)^{1/2}\leq C_0 d^{-1/2} \]

for some constant $C_0>0$ , where the second inequality follows from (4.1). Similarly, Proposition 5.2 and assumption (4.1), applied to the function $x\mapsto (\log(\rho)'(x))^31_{A}(x)$ and random variables $Y^d_i-x^d_i$ , yield

\[ T^d_4(\mathbf{x}^d)\leq C_1 \bigg(\sum_{i=2}^d \,{\mathrm{e}}^{|x^d_i|}/d^3\bigg)^{1/2}\leq C_2 d^{-1} \]

for some constants $C_1,C_2>0$ and all $d\in{\mathbb{N}}$ .

Lemma 4.5. There exist constants $c_1,c'_1>0$ , such that for any $d\in{\mathbb{N}}$ , $i\in\{2,\ldots,d\}$ , $\mathbf{x}^d\in\mathcal{A}_d$ and $x^d_i\notin A$ , we have

\[\bigg|\log\varphi_i (t)-\bigg(i\dfrac{l^2}{2d}\log(\rho)''(x^d_i)\cdot t-\dfrac{l^2}{2d} (\log(\rho)'(x^d_i))^2\cdot t^2\bigg)\bigg|\leq \dfrac{c_1}{d^{3/2}}(t^2+|t|^3),\quad |t|\leq c'_1\sqrt{d},\]

where

\[ \varphi_i(t)\,:\!=\mathbb{E}_{Y_i^d}[\exp(i tK(x^d_i,Y^d_i))],\quad t\in{\mathbb{R}}, \]

is the CF of $K(x^d_i,Y^d_i)$ (see (4.7)).

Proof. By Lemma 5.3, the following inequality holds for all $|t|\leq d/(4l^2|\log(\rho)''(x^d_i)|)$ :

(4.13) \begin{align} & \bigg|\log\varphi_i(t)-\bigg(i\dfrac{l^2}{2d}\log(\rho)''(x^d_i)\cdot t-\dfrac{l^2}{2d} (\log(\rho)'(x^d_i))^2\cdot t^2\bigg)\bigg| \notag \\ &\qquad \leq \dfrac{l^4}{d^2}\bigg(\dfrac{1}{2} (\log(\rho)''(x^d_i))^2\cdot t^2+ (\log(\rho)'(x^d_i))^2 |\log(\rho)''(x^d_i)|\,|t|^3\bigg), \end{align}

Since $\mathbf{x}^d\in\mathcal{A}_d$ and $i>1$ , assumption (4.1) implies that for any $f\in\mathcal{S}^0$ there exists $C_f>0$ such that

\[ | f(x^d_i)|^2/C_f\leq {\mathrm{e}}^{|x^d_i|}\leq \sum_{i=2}^d\,{\mathrm{e}}^{|x^d_i|}\leq 2d\rho({\mathrm{e}}^{|x|}). \]

Hence $| f(x^d_i)|\leq c_f \sqrt{d}$ for all $d\in{\mathbb{N}}$ and all $2\leq i\leq d$ , where $c_f\,:\!=(2C_f\rho({\mathrm{e}}^{|x|}))^{1/2}$ . Since $\log(\rho)\in\mathcal{S}^4$ , both functions

\[ f_1(x)\,:\!=l^4(\log(\rho)''(x))^2/2\quad\text{and}\quad f_2(x)\,:\!=(\log(\rho)'(x))^2|\log(\rho)''(x)| \]

are in $\mathcal{S}^0$ . Then (4.13) and the constants $c_1\,:\!=\max\{c_{f_1},c_{f_2}\}$ and $c_1'\,:\!=1/(4\sqrt{2c_{f_1}})$ yield the inequalities in the lemma.

We now deal with the coordinates of $\mathcal{A}_d$ that are in A. Compared with Lemma 4.5 this is straightforward, as it does not involve the remainder of the coordinates of the point in $\mathcal{A}_d$ .

Moreover, constants $C_1$ and $C_2$ do not depend on the choice of $x\in A$ .

Proof. By definition of A in Proposition 4.1 we have $ |\log(\rho)'(x)|\leq c_A$ and $ |\log(\rho)''(x)|\leq c^2_A$ for $x\in A$ . By (4.7), $\mu_K={l^2}/{(2d)}\log(\rho)''(x)$ and (a) follows. Recall that $\mathbb{E}_{Y}[(Y-x)^n]$ is either zero (if n is odd) or of order $d^{-n/2}$ (if n is even) and $\mathbb{E}_{Y}[(Y-x)^2]=l^2/d$ . Hence the definition of K in (4.7), the fact $x\in A$ , and part (a) imply the inequality in part (b). For part (c), note that an analogous argument yields $\mathbb{E}_{Y} [(K(x,Y)-\mu_K)^6]\leq C'd^{-3}$ for some constant $C'\,{>}\,0$ . Cauchy’s inequality concludes the proof of the lemma.

(4.14) \begin{equation} \bigg|\log\varphi(t)-\bigg(i t\dfrac{l^2}{2d}\log(\rho)''(x)-\dfrac{t^2l^2}{2d}\log(\rho)'(x)^2\bigg)\bigg|\leq c_2\bigg(\dfrac{t^2}{d^2}+\dfrac{|t|^3}{d^{3/2}}+\dfrac{t^4}{d^2}\bigg),\quad |t|\leq c'_2\sqrt{d}. \end{equation}

Proof. Let $\sigma^2_K\,:\!=\mathbb{E}_{Y}[(K(x,Y)-\mu_K)^2]$ and recall $\mu_K={l^2}/{(2d)}\log(\rho)''(x)$ . By Lemma 5.2 we have

(4.15) \begin{equation} \bigg|\log\varphi(t)-\bigg(i t\dfrac{l^2}{2d}\log(\rho)''(x)-\dfrac{t^2}{2}\sigma^2_K\bigg)\bigg| \leq|t|^3\mathbb{E}_{Y} |K(x,Y)-\mu_K|^3/6+t^4\sigma^4_K/4,\quad |t|\leq \dfrac{1}{\sigma_K}. \end{equation}

By Lemma 4.6(b) we have $|\sigma^2_K-l^2\log(\rho)'(x)^2/d|\leq C_1d^{-2}$ . Hence $\sigma^2_K\leq d^{-1}/\sqrt{c_2'}$ , where $c_2'\,:\!= 1/(l^2c_A^2+ C_1)^2$ , and $\sigma^4_K\leq C_1'd^{-2}$ for some $C_1'>0$ . This, together with Lemma 4.6(c), implies that there exists a constant $c_2>0$ , such that the inequality in (4.14) follows from (4.15) for all $|t|\leq c_2' d^{1/2} \leq 1/\sigma_K$ and $x\in A$ .

Lemma 4.8. For any $d\in{\mathbb{N}}$ and $\mathbf{x}^d\in \mathcal{A}_d$ , let $\Phi^d_{K}$ and $\Phi^d_{\mathcal{N}}$ be the distribution functions of $\sum_{i=2}^dK(x^d_i,Y^d_i)$ and $\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)$ . Then there exists $C>0$ such that

\[\sup_{x\in{\mathbb{R}}} |\Phi^d_{\mathcal{N}}(x)-\Phi^d_{K}(x)|\leq C d^{-1/2}\quad \text{for every $d\in{\mathbb{N}}$ and $\mathbf{x}^d\in \mathcal{A}_d$.}\]

Proof. Let $\varphi_K$ and $\varphi_\mathcal{N}$ be the CFs of $\sum_{i=2}^dK(x^d_i,Y^d_i)$ and $\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)$ , respectively. We will compare $\varphi_K$ and $\varphi_\mathcal{N}$ and apply Proposition 5.8 to establish the lemma. Let $\varphi_i$ be the CF of $K(x^d_i,Y^d_i)$ and recall, by (4.9),

\[ \varphi_\mathcal{N}(t)=\exp \bigg(\dfrac{1}{2}it\dfrac{l^2}{d}\sum_{i=2}^d\log(\rho)''(x^d_i)-\dfrac{1}{2}t^2\dfrac{l^2}{d}\sum_{i=2}^d (\log(\rho)'(x^d_i))^2\bigg). \]

Define the positive constants $c\,:\!=\max\{c_1,c_2\}$ and $c'\,:\!=\min\{1,l^2J/(32c),c'_1,c'_2\}$ , where the constants $c_1,c_1'$ (resp. $c_2,c_2'$ ) are given in Lemma 4.5 (resp. Lemma 4.7) and J is as in assumption (4.3). Note that the constants c,c´ do not depend on the choice of $\mathbf{x}^d\in \mathcal{A}_d$ . Lemmas 4.5 and 4.7 imply the following inequality for all $d\in{\mathbb{N}}$ and $\mathbf{x}^d\in \mathcal{A}_d$ :

\begin{equation*} |\log\varphi_K(t)-\log\varphi_\mathcal{N}(t)|\leq \sum_{i=2}^d\bigg|\log\varphi_i(t)-\bigg(i t\dfrac{l^2}{2d}\log(\rho)''(x^d_i)-\dfrac{t^2l^2}{2d} (\log(\rho)(x^d_i)^2\bigg)\bigg| \leq R(t), \end{equation*}

for all $|t|\leq r$ , where $r\,:\!= c'\sqrt{d}$ and $R(t)\,:\!= c(t^2+|t|^3+t^4/\sqrt{d})/\sqrt{d}$ . Since $|t|^3\leq \sqrt{d}c't^2$ and $t^4\leq dc'^2t^2$ for $|t|\leq r$ , we have

(4.16) \begin{equation} R(t)\leq t^2(c/\sqrt{d}+cc'+cc'^2) \leq t^2 (c/\sqrt{d}+2cc')\quad \text{for all $t\in[-r,r]$.} \end{equation}

By assumption (4.3), there exists $d_0'\in{\mathbb{N}}$ such that the variance

\[ \sigma^2_{\mathcal{N}}(\mathbf{x}^d)=l^2/d\sum_{i=2}^d (\log(\rho)'(x^d_i))^2 \]

of $\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)$ satisfies $\sigma^2_{\mathcal{N}}(\mathbf{x}^d)\geq l^2J/2$ for all $d\geq d_0'$ and $\mathbf{x}^d\in \mathcal{A}_d$ . Let $\gamma\,:\!=1/2$ and pick $d_0\,{\in}\,{\mathbb{N}}$ , greater than $\max\{d_0',(16cl^{-2}/J)^2\}$ . Then, for any $d\geq d_0$ , the inequality $c/\sqrt{d}\leq \gamma l^2J/8$ holds. Since $c'\leq l^2J/(32c)$ , we have $2cc'\leq \gamma l^2J/8$ , and the bound in (4.16) implies

\[ R(t)\leq \dfrac{1}{4}t^2 \gamma l^2J \leq \dfrac{1}{2}t^2 \gamma \sigma^2_{\mathcal{N}}(\mathbf{x}^d)\]

for all $t\in[-r,r]$ . By Proposition 5.8, for all $d\geq d_0$ , $\sup_{x\in{\mathbb{R}}} |\Phi^d_{\mathcal{N}}(x)-\Phi^d_{K}(x)|$ is bounded above by

\begin{equation*} \int_{{\mathbb{R}}}\dfrac{R(t)}{\pi |t|}\exp\bigg({-}\dfrac{(1-\gamma)\sigma^2_{\mathcal{N}}(\mathbf{x}^d)t^2}{2}\bigg) \,{\mathrm{d}} t +\dfrac{12\sqrt{2}}{ \pi^{3/2}\sigma_{\mathcal{N}}(\mathbf{x}^d) r}\leq C'/\sqrt{d}, \end{equation*}

where

\[ C'\,:\!= c\int_{\mathbb{R}}(|t|+t^2+|t|^3)\exp({-}l^2Jt^2/8) \,{\mathrm{d}} t + \dfrac{24\sqrt{2}}{\pi^{3/2}l^2Jc'}. \]

Since the left-hand side of the inequality in the lemma is bounded above by 1, the inequality holds for all $d\in{\mathbb{N}}$ if we define $C\,:\!=\max\{C',\sqrt{d_0}\}$ .

Proof of Proposition 4.5. Since $|1\wedge {\mathrm{e}}^y- 1\wedge {\mathrm{e}}^x|\leq |x-y|$ for all $x,y\in{\mathbb{R}}$ , by Lemma 4.4 we have

\[ \Big|\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big]-\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\sum_{i=2}^dK(x^d_i,Y^d_i)}\Big] \Big|\leq C' d^{-1/2}\]

for some constant $C'>0$ and all $d\in{\mathbb{N}}$ . Recall that ${\mathrm{e}}^x 1_{\{x<0\}} = 1\wedge {\mathrm{e}}^x - 1 + 1_{\{x\leq 0\}}$ for all $x\in{\mathbb{R}}\setminus\{0\}$ . Hence Lemmas 4.4 and 4.8 yield

\begin{align*} & \Big|\mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big]-\mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\sum_{i=2}^dK(x^d_i,Y^d_i)}1_{\{\sum_{i=2}^dK(x^d_i,Y^d_i)<0\}}\Big] \Big|\\ &\qquad \leq \Big|\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big]-\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\sum_{i=2}^dK(x^d_i,Y^d_i)}\Big] \Big|+ |\Phi^d_{\mathcal{N}}(0)-\Phi^d_{K}(0)|\\ & \qquad\leq C'' d^{-1/2} \end{align*}

for some $C''>0$ and all $d\in {\mathbb{N}}$ . The proposition follows.

Proof of Proposition 4.6. For any $\mathbf{x}^d\in\mathcal{A}_d$ , by [Reference Roberts, Gelman and Gilks30, Proposition 2.4], we have

\begin{align*} \mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big] & ={\mathrm{e}}^{\mu_{\mathcal{N}}(\mathbf{x}^d)+{\sigma^2_{\mathcal{N}}(\mathbf{x}^d)}/{2}}\Phi \bigg({-}\sigma_{\mathcal{N}}(\mathbf{x}^d)-\frac{\mu_{\mathcal{N}}(\mathbf{x}^d)}{\sigma_{\mathcal{N}}(\mathbf{x}^d)}\bigg), \\ \mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big] & = \mathbb{E}_{\mathbf{Y}^d-} \Big[{\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big] + \Phi\bigg(\frac{\mu_{\mathcal{N}}(\mathbf{x}^d)}{\sigma_{\mathcal{N}}(\mathbf{x}^d)}\bigg). \end{align*}

where $\Phi$ is the distribution of a standard normal random variable. Note first that it is sufficient to prove the inequality in the proposition for all $d> d_0$ for some $d_0\in{\mathbb{N}}$ , since the expectations above are bounded by 1 and we can hence increase the constant C so that the first $d_0$ inequalities are also satisfied.

Recall the formulas for $\mu_{\mathcal{N}}(\mathbf{x}^d)$ and $\sigma^2_{\mathcal{N}}(\mathbf{x}^d)$ from (4.9). By assumptions (4.3) and (4.4) it follows that $ |\mu_{\mathcal{N}}(\mathbf{x}^d)+\sigma^2_{\mathcal{N}}(\mathbf{x}^d)/2|\leq c a_d/\sqrt{d}$ for some constant $c>0$ and all large d and $\mathbf{x}^d\in\mathcal{A}_d$ . Note that $S_a\,:\!=\sup_{d\in{\mathbb{N}}}(a_d/\sqrt{d})<\infty$ since $\{a_d\}_{d\in{\mathbb{N}}}$ is sluggish. The function $x\mapsto {\mathrm{e}}^x$ is Lipschitz on $[-cS_a,cS_a]$ with constant ${\mathrm{e}}^{cS_a}$ . Consequently $ |{\mathrm{e}}^{\mu_{\mathcal{N}}(\mathbf{x}^d)+\sigma^2_{\mathcal{N}}(\mathbf{x}^d)/2}-1|\leq {\mathrm{e}}^{cS_a}a_d/\sqrt{d}$ for large d and uniformly in $\mathbf{x}^d\in\mathcal{A}_d$ .

By assumption (4.3), for all large $d\in{\mathbb{N}}$ and all $\mathbf{x}^d\in\mathcal{A}_d$ , we have $\sigma_{\mathcal{N}}(\mathbf{x}^d)\geq l\sqrt{J}/\sqrt{2}$ . Hence, since the function $x\mapsto \sqrt{x}$ is Lipschitz with constant $c_1\,:\!=1/(l\sqrt{2J})$ on $[{l^2J}/{2},\infty)$ , we obtain

\[ \bigg|\dfrac{\sigma_{\mathcal{N}}(\mathbf{x}^d)}{2}-\dfrac{l\sqrt{J}}{2}\bigg|\leq \dfrac{c_1}{2} |\sigma^2_{\mathcal{N}}(\mathbf{x}^d)-l^2J| \leq c_2 \dfrac{a_d}{\sqrt{d}}, \]

where constant $c_2>0$ exists by (4.3). Moreover,

\[ \bigg|\bigg(\mu_{\mathcal{N}}(\mathbf{x}^d)+\dfrac{\sigma^2_{\mathcal{N}}(\mathbf{x}^d)}{2}\bigg)\cdot\dfrac{1}{\sigma_{\mathcal{N}}(\mathbf{x}^d)}\bigg|\leq c_3 \dfrac{a_d}{\sqrt{d}}\]

for $c_3>0$ and all large d.

Since

\[ \sigma_{\mathcal{N}}(\mathbf{x}^d)+\dfrac{\mu_{\mathcal{N}}(\mathbf{x}^d)}{\sigma_{\mathcal{N}}(\mathbf{x}^d)} =\bigg(\mu_{\mathcal{N}}(\mathbf{x}^d)+ \dfrac{\sigma^2_{\mathcal{N}}(\mathbf{x}^d)}{2}\bigg)\dfrac{1}{\sigma_{\mathcal{N}}(\mathbf{x}^d)}+\dfrac{\sigma_{\mathcal{N}}(\mathbf{x}^d)}{2},\]

the inequalities in the previous paragraph imply that there exists $c_4>0$ such that

\[ \bigg|\sigma_{\mathcal{N}}(\mathbf{x}^d)+\dfrac{\mu_{\mathcal{N}}(\mathbf{x}^d)}{\sigma_{\mathcal{N}}(\mathbf{x}^d)} - \dfrac{l\sqrt{J}}{2}\bigg|\leq c_4 \dfrac{a_d}{\sqrt{d}} \]

for large d and uniformly in $\mathbf{x}^d\in\mathcal{A}_d$ . Since $\Phi$ is Lipschitz with constant $1/\sqrt{2\pi}$ , there exists a constant $C_1'>1$ such that

\[\bigg|\mathbb{E}_{\mathbf{Y}^d-}\Big[\mathrm{e}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}1_{\{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)<0\}}\Big]-\Phi\bigg(\frac{-l\sqrt{J}}{2}\bigg)\bigg|\leq C_1'\frac{a_d}{\sqrt{d}}\]

holds for all large d and all $\mathbf{x}^d\in\mathcal{A}_d$ . Similarly,

\[ \bigg|\mathbb{E}_{\mathbf{Y}^d-} \Big[1\wedge {\mathrm{e}}^{\mathcal{N}(\mathbf{x}^d,\mathbf{Y}^d)}\Big] -2\Phi \bigg(\dfrac{-l\sqrt{J}}{2}\bigg)\bigg|\leq C_2'\dfrac{a_d}{\sqrt{d}} \]

for some $C_2'>0$ all large d and all $\mathbf{x}^d\in\mathcal{A}_d$ , and the proposition follows.

4.3 Proof of Proposition 2.1

We will now prove the following result.

Proposition 4.7. Let $a=\{a_d\}_{d\in{\mathbb{N}}}$ be a sluggish sequence and $p\in[1,\infty)$ . There exists a constant $C_4$ (depending on a and p) such that, for every $f\in\mathcal{S}^3$ and all $d\in{\mathbb{N}}$ , we have

\[\|\mathcal{G}f-{\mathcal{G}}_df\|_p\leq C_4\bigg(\sum_{i=1}^3\|\:f^{(i)}\|_{\infty,1/2}\bigg) \bigg(\dfrac{a_d}{\sqrt{d}} +{\mathrm{e}}^{-a_d^2/p}\bigg).\]

In the case $p=2$ , define $a_d\,:\!=\sqrt{2\log(d)}$ for $d\in{\mathbb{N}}\setminus\{1\}$ and note that Proposition 2.1 then follows as a special case of Proposition 4.7.

Lemma 4.9. There exists a constant C such that, for all $f\in\mathcal{S}^3$ and all $d\in{\mathbb{N}}$ , we have

\[\max\{ |\mathcal{G}f(\mathbf{x}^d)|, |\mathcal{G}_df(\mathbf{x}^d)|\} \leq C \,{\mathrm{e}}^{|x^d_1|}\sum_{i=1}^2\|\:f^{(i)}\|_{\infty,1/2}\quad\text{{for all} } \mathbf{x}^d\in{\mathbb{R}}^d.\]

Proof. The triangle inequality, definition (2.4), and

\[ \log(\rho)'(x)f'(x)\leq \|\log(\rho)'\|_{\infty,1/2}\|\:f'\|_{\infty,1/2}\,{\mathrm{e}}^{|x|} \]

imply the bound in the lemma for $|\mathcal{G}f(\mathbf{x}^d)|$ . To bound $|\mathcal{G}_df(\mathbf{x}^d)|$ , define

\[\tilde{\beta}(\mathbf{x}^d,y)\,:\!=\mathbb{E}_{\mathbf{Y}^d-}\bigg[1\wedge \exp\bigg(\log(\rho)( y)-\log(\rho)(x^d_1)+\sum_{i=2}^d\log(\rho)(Y^d_i)-\log(\rho)(x^d_i)\bigg)\bigg]\]

for any $y\in{\mathbb{R}}$ . Then, if q denotes the density of $Y^d_1-x^d_1\sim N(0,l^2/d)$ , we obtain

(4.17) \begin{align} |\mathcal{G}_df(\mathbf{x}^d)| & =d | \mathbb{E}_{Y^d_1} [ (f(Y^d_1)-f(x^d_1))\tilde{\beta}(\mathbf{x}^d,Y^d_1)]| \notag \\ & \leq d\int_0^{\infty}z | f'(w_1)\tilde{\beta}(\mathbf{x}^d,x^d_1+z)-f'(w_2)\tilde{\beta}(\mathbf{x}^d,x^d_1-z)|q(z) \,{\mathrm{d}} z, \end{align}

where $w_1\in(x_1^d,x_1^d+z)$ and $w_2\in(x_1^d-z,x_1^d)$ satisfy

\[ zf'(w_1)=f(x_1^d+z)-f(x_1^d)\quad\text{and}\quad -zf'(w_2)=f(x_1^d-z)-f(x_1^d), \]

respectively. Moreover, $ | f'(w_1)-f'(w_2)|\leq 2z| f''(w_3)|$ holds for some $w_3$ in the interval ${(x_1^d-z,x_1^d+z)}$ . Since $x\mapsto 1\wedge {\mathrm{e}}^x$ is Lipschitz with constant 1, we obtain

\[ |\tilde{\beta}(\mathbf{x}^d,x^d_1+z)-\tilde{\beta}(\mathbf{x}^d,x^d_1-z)|\leq |\log(\rho)(x_1^d+z)-\log(\rho)(x_1^d-z)| \leq 2z|\log(\rho)'(w_4)|\]

for some $w_4\in (x_1^d-z,x_1^d+z)$ . By adding and subtracting $f'(w_2)\tilde{\beta}(\mathbf{x}^d,x^d_1+z)$ on the right-hand side of (4.17), applying the two bounds we just derived and noting that $\tilde \beta\leq 1$ , we obtain

(4.18) \begin{equation} |\mathcal{G}_df(\mathbf{x}^d)| \leq 2d\int_0^\infty z^2| f''(w_3)|q(z) \,{\mathrm{d}} z+ 2d\int_0^\infty z^2| f'(w_2)\log(\rho)'(w_4)|q(z) \,{\mathrm{d}} z. \end{equation}

Note that, since $\max\{|w_3|,|w_2|,|w_4|\} \leq |x_1^d|+z$ and $\|\:f''\|_{\infty,1}\leq \|\:f''\|_{\infty,1/2}$ , we have

\begin{gather*} | f''(w_3)| \leq \|\:f''\|_{\infty,1} \,{\mathrm{e}}^{|w_3|} \leq \|\:f''\|_{\infty,1/2} \,{\mathrm{e}}^{|x_1^d|+z},\\ \log(\rho)'(w_4)f'(w_2) \leq \|\log(\rho)'\|_{\infty,1/2}\|\:f'\|_{\infty,1/2}\,{\mathrm{e}}^{|x_1^d|+z}, \end{gather*}

which, together with inequality (4.18), implies the lemma.

Proof of Proposition 4.7. By Theorem 2.2 (on $\mathcal{A}_d$ ) and Lemma 4.9 (on ${\mathbb{R}}^d\setminus \mathcal{A}_d$ ), there exists a constant $C>0$ such that for any $f\in\mathcal{S}^3$ the following inequality holds:

\begin{align*} \|\mathcal{G}_df-\mathcal{G}f\|^p_p & = \int_{\mathcal{A}_d} |\mathcal{G}_df(\mathbf{x}^d)-\mathcal{G}f(\mathbf{x}^d)|^p\rho_d(\mathbf{x}^d) \,{\mathrm{d}} \mathbf{x}^d \\ &\quad\, + \int_{{\mathbb{R}}^d\setminus \mathcal{A}_d} |\mathcal{G}_df(\mathbf{x}^d)-\mathcal{G}f(\mathbf{x}^d)|^p\rho_d(\mathbf{x}^d) \,{\mathrm{d}} \mathbf{x}^d\\ & \leq C\rho({\mathrm{e}}^{p|x|})\bigg(\dfrac{a_d^p}{d^{p/2}}\rho_d(\mathcal{A}_d)+ \rho_d({\mathbb{R}}^d\setminus \mathcal{A}_d)\bigg)\bigg(\sum_{i=1}^3\|\:f^{(i)}\|_{\infty,1/2}\bigg)^p. \end{align*}

Apply Proposition 4.2 and raise both sides of the inequality to the power $1/p$ to conclude the proof of the proposition.

4.4 Proof of Theorem 2.1

Proof. The lemma follows from [Reference Jarner and Hansen16, Theorem 4.1] if we prove that the target $\rho_d$ satisfies

(4.19) \begin{gather} \lim_{|\mathbf{x}^d|\to\infty}\dfrac{\mathbf{x}^d}{|\mathbf{x}^d|}\cdot\nabla\log(\rho_d(\mathbf{x}^d))= \lim_{|\mathbf{x}^d|\to\infty}\sum_{i=1}^d\dfrac{x^d_i}{|x^d_i|}\log(\rho)'(x^d_i)=-\infty,\end{gather}

(4.20) \begin{gather}* \liminf_{|\mathbf{x}^d|\to\infty} \mathbb{P}_{\mathbf{Y}^d} [\rho_d(\mathbf{Y}^d)\geq \rho_d(\mathbf{x}^d)]>0. \end{gather}

Assumption (2.3) implies that the expression $x/|x|\cdot \log(\rho)'(x)$ is bounded above and takes arbitrarily large negative values as $|x|\to\infty$ . This yields (4.19), since $|\mathbf{x}^d|\to\infty$ implies that $|x^d_i|\to\infty$ holds for at least one $i\in\{1,\ldots,d\}$ .

Condition (4.20) states that the acceptance probability in the RWM chain is bounded away from zero sufficiently far from the origin. To prove this, recall that $\mathbf{Y}^d\sim N(\mathbf{x}^d,l^2/d\cdot I_d)$ and define the set

\[B(\mathbf{x}^d)\,:\!=\bigg\{\mathbf{y}^d\in{\mathbb{R}}^d\colon \dfrac{x^d_i}{|x^d_i|}\cdot( y^d_i-x^d_i)\in \bigg(\dfrac{-2l}{\sqrt{d}},\dfrac{-l}{\sqrt{d}}\bigg)\text{ for all }i\leq d\bigg\},\]

where we interpret $x^d_i/|x^d_i|\,:\!=1$ if $x^d_i=0$ . Clearly $\inf_{\mathbf{x}^d\in{\mathbb{R}}^d}\mathbb{P}_{\mathbf{Y}^d} [B(\mathbf{x}^d) ]>0$ . We now prove that if $|\mathbf{x}^d|$ is sufficiently large, then $\rho_d(\mathbf{y}^d)\geq\rho_d(\mathbf{x}^d)$ for all $\mathbf{y}^d\in B(\mathbf{x}^d)$ , which implies (4.20).

By (2.3), far enough from zero, $\rho$ is decreasing in a direction away from the origin. Therefore, there exists a compact interval $K\subset {\mathbb{R}}$ such that $ ({-}2l/\sqrt{d},2l/\sqrt{d})\subset K $ and $\rho( y)\geq \rho(x)$ whenever $x\notin K$ and $x/|x|\cdot ( y-x)\in ({-}2l/\sqrt{d},-l/\sqrt{d})$ . We claim that for every $\mathbf{y}^d\in B(\mathbf{x}^d)$ , the inequality $\rho( y^d_i)/\rho(x^d_i)\geq (\min_{x\in K}\rho(x))/(\max_{x\in{\mathbb{R}}}\rho(x))\in(0,1)$ holds. If $x^d_i\in K$ , then $y^d_i\in K$ and the inequality follows trivially. If $x^d_i\notin K$ , then, by the definition of K, we have $\rho( y^d_i)/\rho(x^d_i)\geq 1$ . This proves the claim. Hence, for $\mathbf{y}^d\in B(\mathbf{x}^d)$ we have

(4.21) \begin{equation} \dfrac{\rho_d(\mathbf{y}^d)}{\rho_d(\mathbf{x}^d)}\geq \bigg(\max_{i\leq d}\dfrac{\rho( y^d_i)}{\rho(x^d_i)}\bigg)\cdot\bigg(\dfrac{\min_{x\in K}\rho(x)}{\max_{x\in{\mathbb{R}}}\rho(x)}\bigg)^{d-1}. \end{equation}

We now prove that the ratio $\rho( y^d_i)/\rho(x^d_i)$ takes arbitrarily large values as $|x^d_i|\to\infty$ . To show this, pick $\mathbf{y}^d\in B(\mathbf{x}^d)$ and assume the inequality $y_i^d>x_i^d$ . Then $x_i^d<0$ and $y_i^d-x_i^d>l/\sqrt{d}$ .

Moreover, the following holds:

\[\dfrac{\rho( y^d_i)}{\rho(x^d_i)}=\exp\bigg(\log\bigg(\dfrac{\rho( y^d_i)}{\rho(x^d_i)}\bigg)\bigg)\geq 1+\int_{x^d_i}^{y^d_i}\!\log(\rho)'(z) \,{\mathrm{d}} z\geq 1+l/\sqrt{d}\inf_{z<x^d_i+2l/\sqrt{d}}\log(\rho)'(z)\to\infty\]

as $x_i^d\to-\infty$ by (2.3). This, together with (4.21), implies (4.20). The case $y_i^d<x_i^d$ is analogous and the lemma follows.

Proof. Clearly, if $f\in\mathcal{C}^{n_f}$ and $\rho\in\mathcal{C}^{n_\rho}$ and if $\rho$ is strictly positive, then $\skew3\hat{f}\in\mathcal{C}^{\min(n_f+2,n_\rho+1)}$ . Pick $s>0$ . L’Hôpital’s rule implies

\[\lim_{x\to\infty}\dfrac{\skew3\hat{f}(x)}{{\mathrm{e}}^{s|x|}}= \dfrac{2}{sh(l)}\lim_{x\to\infty}\dfrac{\int_{-\infty}^x\rho( y)(\rho(f)-f( y)) \,{\mathrm{d}} y}{{\mathrm{e}}^{sx}\rho(x)}=\dfrac{2}{sh(l)}\lim_{x\to\infty}\dfrac{\rho(f)-f(x)}{s\,{\mathrm{e}}^{sx}+{\mathrm{e}}^{sx}(\log(\rho))'(x)}.\]

The last limit is zero by (2.3). An analogous argument shows $\lim_{x\to-\infty}\skew3\hat{f}(x)/{\mathrm{e}}^{s|x|}=0$ . Hence $\|\skew3\hat{f}\|_{\infty,s}<\infty$ holds for all $s>0$ . Since

\[ \dfrac{h(l)}{2}\hat{f}'(x)=\dfrac{1}{\rho(x)}\bigg(\int_{-\infty}^x\rho(y)(\rho(f)-f(y))\,\mathrm{d}y\bigg), \]

this argument implies that $\|\skew3\hat{f}'\|_{\infty,s}<\infty$ holds for all $s>0$ . Hence $\skew3\hat{f}\in \mathcal{S}^{1}$ .

Proceed by induction. Assume that for all $k\leq n$ (where $1\leq n<\min(n_f+2,n_\rho+1)$ ) we have $\|\skew3\hat{f}^{(k)}\|_{\infty,s}<\infty$ for any $s>0$ . Pick an arbitrary $u>0$ . By differentiating (2.5) we obtain

\[\skew3\hat{f}^{(n+1)}=-\sum_{k=0}^{n-1}\binom{n-1}{k}(\log(\rho))^{(k+1)}\skew3\hat{f}^{(n-k)}+\dfrac{2}{h(l)}(\rho(f)-f)^{(n-1)}.\]

Since $n\leq \min(n_\rho, n_f+1)$ , the induction hypothesis implies $\|\skew3\hat{f}^{(k)}\|_{\infty,u/2}<\infty$ for all $1\,{\leq}\, k\,{\leq}\,n$ . By assumption we have $\|\:f^{(n-1)}\|_{\infty,u}<\infty$ and $\|(\log(\rho))^{(k)}\|_{\infty,u/2}<\infty$ for all $1\leq k\leq n$ . Hence $\|\skew3\hat{f}^{(n+1)}\|_{\infty,u}<\infty$ holds for an arbitrary $u>0$ and the proposition follows.

Proof of Theorem 2.1. By Lemma 4.10, the RWM chain $\mathbf{X}^d$ with the transition kernel $P_d$ is V-uniformly ergodic with $V=\rho_d^{-1/2}$ . Moreover, by [Reference Roberts and Rosenthal27, Proposition 2.1 and Theorem 2.1], $P_d$ defines a self-adjoint operator on $\{g\in L^2(\rho_d)\colon \rho_d(g)=0\}$ with norm $\lambda_d<1$ . Proposition 4.4 implies $\skew3\hat{f}\in\mathcal{S}^3$ , since by assumption we have $f\in\mathcal{S}^1$ and $\log(\rho)\in\mathcal{S}^4$ . By Remark 5.1(c) in Section 5 below we have $\skew3\hat{f}^2\in\mathcal{S}^3$ . Since $P_d\skew3\hat f = (1/d)\mathcal{G}_d\skew3\hat f+\skew3\hat f$ , Lemma 4.3 implies that $ (P_d\skew3\hat{f})^2(\mathbf{x}^d)\leq C_{\skew3\hat{f}} \,{\mathrm{e}}^{2 |x_1^d|} $ for some positive constant $C_{\skew3\hat{f}}$ and all $\mathbf{x}^d\in{\mathbb{R}}^d$ . Hence (2.3) and the definition of V imply the inequality $\max\{\skew3\hat{f}^2,(P_d\skew3\hat{f})^2\}\leq cV$ for some constant $c>0$ . Consequently, by [Reference Meyn and Tweedie22, Theorem 17.0.1], the CLT for the chain $\mathbf{X}^d$ and function $f+dP_d\skew3\hat{f}-d\skew3\hat{f}$ holds with some asymptotic variance $\hat \sigma^2_{f,d}$ .

By [Reference Kipnis and Varadhan18] and [Reference Geyer14] we can represent $\hat \sigma^2_{f,d}$ in terms of a positive spectral measure $E_d( {\mathrm{d}} \lambda)$ associated with the function $f-\rho(f)+dP_d\skew3\hat{f}-d\skew3\hat{f}=\mathcal{G}_d\skew3\hat{f}-\mathcal{G}\skew3\hat{f}$ as

\[\hat \sigma^2_{f,d}=\int_{\Lambda_d}\dfrac{1+\lambda}{1-\lambda}E_d({\mathrm{d}} \lambda),\]

where $\Lambda_d\subset[-\lambda_d,\lambda_d]$ denotes the spectrum of the self-adjoint operator $P_d$ acting on the Hilbert space $\{g\in L^2(\rho_d)\colon \rho_d(g)=0\}$ . By the definition of the spectral measure $E_d( {\mathrm{d}} \lambda)$ , we obtain the bound

\[\hat \sigma^2_{f,d}\leq\dfrac{1+\lambda_d}{1-\lambda_d}\int_{\Lambda_d}E_d( {\mathrm{d}} \lambda)=\dfrac{1+\lambda_d}{1-\lambda_d} \|P_d(d\skew3\hat{f})-d\skew3\hat{f}+f-\rho(f)\|^2_2\leq \dfrac{2}{1-\lambda_d}\|\mathcal{G}_d\skew3\hat{f}-\mathcal{G}\skew3\hat{f}\|^2_2.\]

Finally, the result follows by Proposition 2.1.

5.1. Bounds on the expectations of test functions

We start with elementary observations.

Proposition 5.1. Pick an arbitrary $n\in{\mathbb{N}}$ . Assume $f\in\mathcal{S}^n$ , $k\leq n$ , $x\in{\mathbb{R}}$ and $Y\sim N(x,\sigma^2)$ . Then there exists measurable Z satisfying

\[ \dfrac{1}{k!}f^{(k)}(Z)(Y-x)^k=f(Y)-\sum_{i=0}^{k-1}\dfrac{1}{i!}f^{(i)}(x)(Y-x)^i \]

and $|Z-x|<|Y-x|$ . Furthermore, there exists a constant $C>0$ (depending on n) such that, for any $m\in{\mathbb{N}}$ and $s>0$ , we have

\[\mathbb{E}_{Y} [ | f^{(k)}(Z)|^{m} |Y-x|^{n}]\leq C\,{\mathrm{e}}^{s^2\sigma^2}\mathbb{E}_{Y} [ |Y-x|^{n}]\|\:f^{(k)}\|_{\infty,s/m}^m\,{\mathrm{e}}^{s|x|}.\]

Proof. A random variable Z, defined via the integral form of the remainder in Taylor’s theorem, lies almost surely between Y and x, implying $|Z-x|<|Y-x|$ . Cauchy’s inequality yields

(5.1) \begin{equation} \mathbb{E}_{Y} [ | f^{(k)}(Z)|^{m} |Y-x|^{n}]^2\leq \mathbb{E}_{Y} [ | f^{(k)}(Z)|^{2m}]\mathbb{E}_{Y} [ |Y-x|^{2n}]. \end{equation}

Since $f\in\mathcal{S}^n\subset\mathcal{S}^k$ , we have

\[ \sup_{x\in{\mathbb{R}}} | f^{(k)}(x)|^{2m}\,{\mathrm{e}}^{-2s|x|}=\|\:f^{(k)}\|^{2m}_{\infty,s/m}<\infty. \]

As $Y\sim N(x,\sigma^2)$ , we have the equality

\[ \mathbb{E}_{Y} [ |Y-x|^{2n}]= C^2 \mathbb{E}_{Y} [ |Y-x|^{n}]^2, \]

where

\[ C\,:\!=\dfrac{\pi^{1/4}\,\Gamma(n+{1}/{2})^{1/2}}{\Gamma({(n+1)}/{2})} \]

and $\Gamma(\cdot)$ is the Euler gamma function. Hence, by (5.1), we obtain

\[\mathbb{E}_{Y} [ | f^{(k)}(Z)|^{m} |Y-x|^{n}]\leq \dfrac{C}{\sqrt{2}}\|\:f^{(k)}\|^{m}_{\infty,s/m}\sqrt{\mathbb{E}_{Y} [{\mathrm{e}}^{2s|Z|}]}\mathbb{E}_{Y} [ |Y-x|^{n}].\]

It remains to note that

\begin{equation*} \mathbb{E}_{Y}\,{\mathrm{e}}^{2s(|Z|-|x|)} \leq \mathbb{E}_{Y}\,{\mathrm{e}}^{2s|Z-x|} \leq \mathbb{E}_{Y}\,{\mathrm{e}}^{2s|Y-x|} \leq2 \,\mathbb{E}_{Y}\,{\mathrm{e}}^{2s(Y-x)}= 2\,{\mathrm{e}}^{2s^2\sigma^2}. \end{equation*}

Proposition 5.2. Let $f\colon{\mathbb{R}}\to{\mathbb{R}}$ be a measurable (not necessarily continuous) function such that $\|\:f\|_{\infty,1/2}<\infty$ . Fix $n\in{\mathbb{N}}$ , $\mathbf{x}^d\in{\mathbb{R}}^d$ and let $X_1, X_2\ldots, X_d$ be i.i.d. copies of X, satisfying $\mathbb{E} [X^n]=0$ and $\mathbb{E} [X^{2n}]<\infty$ . Then the following inequality holds:

\[\bigg|\mathbb{E}\bigg[\sum_{i=1}^df(x^d_i)X_i^n\bigg]\bigg|\leq \|\:f\|_{\infty,1/2} \bigg(\mathbb{E}[X^{2n}]\sum_{i=1}^d\,{\mathrm{e}}^{|x^d_i|}\bigg)^{1/2}.\]

Proof. By Jensen’s inequality, the fact that $\mathbb{E}[X]=0$ , and the assumption on f, we obtain

\begin{equation*} \mathbb{E}\bigg[\sum_{i=1}^df(x^d_i)X_i^n\bigg]^2\leq \mathbb{E}\bigg[\bigg(\sum_{i=1}^df(x^d_i)X_i^n\bigg)^2\bigg]= \sum_{i=1}^d (f(x^d_i))^2\mathbb{E} [X_i^{2n}]\leq\|\:f\|^{2}_{\infty,1/2} \mathbb{E}[X^{2n}]\sum_{i=1}^d\,{\mathrm{e}}^{|x^d_i|}. \end{equation*}

5.2. Deviations of the sums of i.i.d. random variables

Proposition 5.3. Let $f\in \mathcal{S}^0$ be such that $\rho(f)=0$ and let $a=\{a_d\}_{d\in{\mathbb{N}}}$ be a sluggish sequence. If the random vector $(X_{1,d},\ldots, X_{d,d})$ follows the density $\rho_d$ for all $d\in{\mathbb{N}}$ , then for every $t>0$ the following inequality holds for all but finitely many $d\in{\mathbb{N}}$ :

\[ \mathbb{P}_{\rho_d}\bigg[\bigg|\dfrac{1}{d-1}\sum_{i=2}^df(X_{i,d})\bigg|\geq \dfrac{ta_d}{\sqrt{d}}\bigg]\leq \exp\bigg(-\dfrac{t^2a_d^2}{3\rho(f^2)}\bigg). \]

Remark 5.3. Proposition 5.3 is an elementary consequence of a deeper underlying result, that the sequence of random variables

\[ \bigg\{\dfrac{1}{a_d\sqrt{d}}\sum_{i=1}^df(X_{i,d})\bigg\}_{d\in\mathbb{N}} \]

satisfies a moderate deviation principle with a good rate function $t\mapsto t^2/(2\rho(f^2))$ and speed $a_d^2$ (see [Reference Eichelsbacher and Löwe13] for details). The key inequality needed in the proof of Proposition 5.3 is given in the next lemma.

Lemma 5.1. Let the assumptions of Proposition 5.3 hold. If $\rho(f^2)>0$ , then for every closed $F\subseteq {\mathbb{R}}$ the following holds:

\[\limsup_{d\to\infty}a^{-2}_d\log\mathbb{P}_{\rho_d}\bigg[\dfrac{1}{a_d\sqrt{d}}\sum_{i=1}^df(X_{i,d})\in F\bigg]\leq -\inf\bigg\{ \dfrac{x^2}{2\rho(f^2)}\colon x\in F\bigg\}. \]

Proof. The moderate deviations results [Reference Eichelsbacher and Löwe13, Theorem 2.2, Lemma 2.5, Remark 2.6] yield a sufficient condition for the above inequality. More precisely, for $X\sim \rho$ , we need to establish

(5.2) \begin{equation} \limsup_{d\to\infty} a_d^{-2}\log (d\cdot \mathbb{P}_{\rho} [| f(X)|\geq a_d\sqrt{d}])=-\infty. \end{equation}

Fix an arbitrary $m\in{\mathbb{N}}$ . Since $f\in\mathcal{S}^0$ , we have $| f(x)|\leq \|\:f\|_{\infty,1/m} \,{\mathrm{e}}^{|x|/m}$ for every $x\in{\mathbb{R}}$ . Consequently, for all large d, we obtain

\[\mathbb{P}_{\rho} [| f(X)|\geq a_d\sqrt{d}]\leq \mathbb{P}_{\rho} [\|\:f\|_{\infty,1/m}^m \,{\mathrm{e}}^{|X|}\geq d^{m/2}]\leq \|\:f\|_{\infty,1/m}^m \rho ({\mathrm{e}}^{|X|})d^{-m/2}.\]

Since $\{a_d\}_{d\in{\mathbb{N}}}$ is sluggish, there exists $ C_0>0$ such that

\[ a_d^{-2}\log(\|\:f\|_{\infty,1/m}^m \rho({\mathrm{e}}^{|X|})){<}C_0{<} a_d^{-2}\log(d) \]

for all large $d\in{\mathbb{N}}$ . Hence

\begin{align*} a_d^{-2}\log (d\cdot\mathbb{P}_{\rho}[| f(X)| \geq a_d\sqrt{d}]) & \leq a_d^{-2}(\log(\|\:f\|_{\infty,1/m}^m \rho({\mathrm{e}}^{|X|}))-(m/2-1)\log(d))\\ &< -C_0(m/2-2), \end{align*}

for all large $d\in{\mathbb{N}}$ . Since m was arbitrary, (5.2) follows.

Proof of Proposition 5.3. Note that the proposition holds if $\rho(f^2)=0$ . Assume $\rho(f^2)\,{>}\,0$ and fix an arbitrary $t>0$ . Note that since $\{a_d\}_{d\in{\mathbb{N}}}$ is sluggish, so is $\{a'_d\}_{d\in{\mathbb{N}}}$ , $a'_d\,:\!=a_{d+1}\sqrt{d/(d+1)}$ . Apply Lemma 5.1 to $F={\mathbb{R}}\setminus ({-}t,t)$ and $\{a'_d\}_{d\in{\mathbb{N}}}$ to get the following inequality:

(5.3) \begin{equation} \mathbb{P}_{\rho_{d-1}}\bigg[\bigg|\dfrac{1}{a'_{d-1}\sqrt{d-1}}\sum_{i=1}^{d-1}f(X_{i,d-1})\bigg|\geq t\bigg]\leq \exp\bigg(-\dfrac{3(a'_{d-1})^2t^2}{8\rho(f^2)}\bigg) \end{equation}

for all sufficiently large $d\in{\mathbb{N}}$ . Since $3(a'_{d-1})^2/4\geq 2a^2_d/3$ for all but finitely many $d\in{\mathbb{N}}$ , the right-hand side of (5.3) is bounded above by $\exp({-}(a_d)^2t^2/(3\rho(f^2)))$ . Recall that $\rho_d(\mathbf{x}^d)=\rho_{d-1}(\mathbf{x}^{d-1})\rho(x_d^d)$ and $a_{d-1}'\sqrt{d-1}=a_d(d-1)/\sqrt{d}$ . Hence the left-hand side of inequality (5.3) equals

\[ \mathbb{P}_{\rho_d}\bigg[\bigg|\dfrac{1}{d-1}\sum_{i=2}^{d}f(X_{i,d})\bigg|\geq t\dfrac{a_d}{\sqrt{d}}\bigg] \]

and the proposition follows.

The next result is based on a combinatorial argument. A special case of Proposition 5.4 was used in [Reference Roberts, Gelman and Gilks30].

Proposition 5.4. Let $n\in{\mathbb{N}}$ and a measurable $f\colon{\mathbb{R}}\to{\mathbb{R}}$ satisfy $\rho(f)=0$ and $\rho(f^{2n})< \infty$ . If the random vector $(X_{1,d},\ldots,X_{d,d})$ is distributed according to $\rho_d$ , then there exists a constant C, independent of d, such that

\[ \mathbb{P}_{\rho_d} \bigg[ \bigg|\dfrac{1}{d-1}\sum_{i=2}^df(X_{i,d})\bigg|\geq 1\bigg]\leq C d^{-n}. \]

Proof of Proposition 5.4. Fix $n\in{\mathbb{N}}$ and let ${\mathbb{N}}_0\,:\!={\mathbb{N}}\cup\{0\}$ . Markov’s inequality and the multinomial theorem yield

\begin{align*} \mathbb{P}_{\rho_d}\bigg[\bigg|\dfrac{1}{d-1}\sum_{i=2}^df(X_{i,d})\bigg|\geq 1\bigg] & =\mathbb{P}_{\rho_d}\bigg[\bigg|\dfrac{1}{d-1}\sum_{i=2}^d f(X_{i,d})\bigg|^{2n}\geq 1\bigg] \\ & \leq \mathbb{E}_{\rho_d}\bigg(\dfrac{1}{d-1}\sum_{i=2}^d f(X_{i,d})\bigg)^{2n} \\ & =(d-1)^{-2n}\sum_{\substack{k_2+k_3+\cdots +k_d=2n\\k_2,k_3\ldots,k_d\in{\mathbb{N}}_0\setminus \{1\}}}\binom{2n}{k_2,k_3,\ldots,k_d}\prod_{i=2}^d\mathbb{E}_{\rho} [ f(X_{i,d})^{k_i}], \end{align*}

where the last equality holds since the expectation of any summand of the form $\prod_{i=2}^{d}f(X_{i,d})^{k_i}$ is zero if any of the indices $k_i=1$ since $\rho_d$ has a product structure and $\rho(f)=0$ . By Jensen’s inequality,

\[ \prod_{i=2}^d\mathbb{E}_{\rho} [ f(X_{i,d})^{k_i}]\leq \prod_{i=2}^d\mathbb{E}_{\rho} [ f(X_{i,d})^{2n}]^{k_i/2n}=\rho(f^{2n}) ^{\sum_{i=2}^{d}{k_i}/{2n}}=\rho(f^{2n}), \]

and hence

(5.4) \begin{equation} \mathbb{P}_{\rho_d}\bigg[\bigg|\dfrac{1}{d-1}\sum_{i=2}^df(X_{i,d})\bigg|\geq 1\bigg]\leq (2n)!\cdot\rho(f^{2n})(d-1)^{-2n}\cdot |\mathcal{N}_d|, \end{equation}

where $ |\mathcal{N}_d|$ stands for the cardinality of the set

\[\mathcal{N}_d\,:\!=\bigg\{(k_2,k_3,\ldots, k_d)\in{\mathbb{N}}^{d-1}_0\colon \sum_{i=2}^dk_d=2n\text{ and } k_i\neq 1\text{ for all } 2\leq i\leq d\bigg\}.\]

Inequality (5.4) and the next claim prove the proposition.

Claim. $ |\mathcal{N}_d|\leq C' d^n$ for a constant C’ independent of d.

Proof of the claim. Consider a function

\[ \zeta\colon {\mathbb{N}}^{d-1}_0\to{\mathbb{N}}^{d-1}_0,\quad \zeta(a_2,a_3,\ldots, a_d)\,:\!=\bigg(2\bigg\lfloor\dfrac{a_2}{2}\bigg\rfloor,2\bigg\lfloor\dfrac{a_3}{2}\bigg\rfloor,\ldots, 2\bigg\lfloor\dfrac{a_d}{2}\bigg\rfloor \bigg), \]

that rounds each entry down to the nearest even number. Every element in the image $\zeta(\mathcal{N}_d)$ is a $(d-1)$ -tuple of non-negative even integers with sum at most 2n. Recall that the number of k-combinations with repetition, chosen from a set of $d-1$ objects, equals $\binom{k+d-2}{k}$ . There exists $C''>0$ , such that

\begin{align*} |\zeta(\mathcal{N}_d)|&\leq \bigg|\bigg\{(k_2,k_3,\ldots, k_d)\in{\mathbb{N}}^{d-1}_0\colon \sum_{i=2}^dk_d\leq n\bigg\}\bigg| \\ &= \sum_{k=0}^n \bigg|\bigg\{(k_2,k_3,\ldots, k_d)\in{\mathbb{N}}^{d-1}_0\colon \sum_{i=2}^dk_d=k\bigg\}\bigg|\\ & =\sum_{k=0}^n\binom{k+d-2}{k}\\ & \leq C'' d^n. \end{align*}

Note that the pre-image of a singleton under $\zeta$ contains at most $2^n$ elements (i.e. $(d-1)$ -tuples) of $\mathcal{N}_d$ . Indeed, by the definition of $\mathcal{N}_d$ , at most n coordinates of an element are not zero and each can either reduce by one or stay the same. Hence, for $C'\,:\!=C'' 2^n$ , we have $ |\mathcal{N}_d|\leq 2^n\cdot |\zeta(\mathcal{N}_d)|\leq C'd^n$ .

5.3. Bounds on the densities of certain random variables

The key step in the proof of Proposition 5.5 below is the optimal Young’s inequality: for $p,q\geq 1$ and $r\in[1,\infty]$ , such that $1/p+1/q=1+1/r$ , and functions $f\in L^p({\mathbb{R}})$ and $g\in L^q({\mathbb{R}})$ , their convolution $f*g$ satisfies the inequality

(5.5) \begin{equation} \|\:f*g\|_r\leq \dfrac{C_pC_q}{C_r}\|\:f\|_p\|g\|_q,\quad\text{where } C_s\,:\!=\begin{cases} \sqrt{\dfrac{s^{1/s}}{s'^{1/s'}}}& \text{if $s\in(1,\infty)$ and $1/s+1/s'=1$,}\\ 1& \text{if $s\in\{1,\infty\}$.} \end{cases} \end{equation}

For $s\in[1,\infty)$ , $\|\cdot\|_s$ is the usual norm on $L^s({\mathbb{R}})$ and $\|\cdot\|_\infty$ denotes the essential supremum norm on $L^\infty({\mathbb{R}})$ . The proof of (5.5) for $r<\infty$ is given in [Reference Barthe2, Theorem 1]. In the case $r=\infty$ , we have $C_pC_q/C_r=1$ and the inequality in (5.5) follows from the definition of the convolution, translation invariance of the Lebesgue measure and Hölder’s inequality.

Proof of Proposition 5.5. Since random variables $X_i$ are independent, the density of their sum is a convolution of the respective densities, $Q_d=\mathop{\mbox{\ast}}_{i=1}^d q_i$ . For all i and each $t>1$ we have $q_i\in L^{\infty}({\mathbb{R}})\cap L^{1}({\mathbb{R}})\subset L^{t}({\mathbb{R}})$ . Moreover, the following inequality holds for every $k\leq d-1$ :

(5.6) \begin{equation} \|Q_d\|_{\infty}=\bigg\|\mathop{\mbox{\ast}}_{i=1}^d q_i\bigg\|_{\infty}\leq (C_{{d}/{d-1}})^kC_{{d}/{k}} \bigg(\prod_{i=1}^k\|q_i\|_{{d}{d-1}}\bigg)\bigg\|\mathop{\mbox{\ast}}_{i=k+1}^d q_i\bigg\|_{{d}/{k}} \end{equation}

We prove (5.6) by induction on k. For $k=1$ , note that d and ${d}/{d-1}$ are Hölder conjugates, i.e. $1/d +1/(d/(d-1))=1$ . Hence (5.5) for $r=\infty$ , $q=d$ , $p=d/(d-1)$ , $f=q_1$ and $g\,{=}\,\mathop{\mbox{\ast}}_{i=2}^d q_i$ implies $\|Q_d\|_{\infty}\leq \|q_1\|_{{d}/{d-1}}\|\mathop{\mbox{\ast}}_{i=2}^d q_i\|_{d}$ and $C_{{d}/{d-1}}=C^{-1}_d$ . Now assume (5.6) holds for some $k\leq d-2$ . Since $ (d/(d-1))^{-1}+ (d/(k+1))^{-1}=1+ (d/k)^{-1}$ , the inequality in (5.5) implies

\[\bigg\|\mathop{\mbox{\ast}}_{i=k+1}^d q_i\bigg\|_{{d}/{k}}\leq \frac{C_{{d}/{d-1}}C_{{d}/{k+1}}}{C_{{d}/{k}}}\|q_{k+1}\|_{{d}/{d-1}}\bigg\|\mathop{\mbox{\ast}}_{i=k+2}^d q_i\bigg\|_{{d}/{k+1}}.\]

This inequality and the induction hypothesis (i.e. (5.6) for k) implies (5.6) for $k+1$ .

Since $q_1$ is a density, we have $\|q_i\|_1=1$ . Hence we find

\[ \|q_i\|_{{d}/{d-1}}\leq \|q_i\|_1^{{d-1}/{d}}\|q_i\|_{\infty}^{{1}/{d}}=\|q_i\|_{\infty}^{{1}/{d}} \]

for each i, and in particular $\prod_{i=1}^d\|q_i\|_{{d}/{d-1}}\leq \max_{i\leq d} (\|q_i\|_\infty)$ . By (5.6), for $k=d-1$ we obtain

\[\|Q_d\|_{\infty}\leq (C_{{d}/{d-1}})^d\prod_{i=1}^d\|q_i\|_{{d}/{d-1}}\leq \max_{i\leq d} (\|q_i\|_\infty) (C_{{d}/{d-1}})^d.\]

Since $\lim_{d\to\infty}\sqrt{d} (C_{{d}/{d-1}})^d=\sqrt{e}$ , there exists $c>0$ such that $ (C_{{d}/{d-1}})^d\leq c/ \sqrt{d}$ for all $d\in{\mathbb{N}}$ .

Polynomials of continuous random variables play an important role in the proofs of Section 4.

Proof. The set $B\,:\!=p ((p')^{-1} (\{0\}))$ has finitely many points. Moreover, p is locally invertible on ${\mathbb{R}}\setminus B$ by the inverse function theorem and the inverses are differentiable. Hence, for any $x\,{\notin}\, B$ , the set $p^{-1} (({-}\infty,x])$ is a disjoint union of intervals with boundaries that depend smoothly on x. Since $\mathbb{P}[p(X)\leq x]=\mathbb{P}[X\in p^{-1} (({-}\infty,x])]$ , the proposition follows.

Let $N=N(\mu,\sigma^2)$ be a normal random variable and let p be a polynomialatisfying $\inf_{x\in{\mathbb{R}}}|p'(x)|\geq c_p$ for some constant $c_p>0$ . Then the random variable p(N) has a probability density function $q_{p(N)}$ , which satisfies $\|q_{p(N)}\|_\infty\leq (c_p\sigma\sqrt{2\pi})^{-1}$ .

Proof. Obviously, p is strictly monotonic and thus a bijection. Moreover, the distribution $\Phi_{p(N)}(\cdot)$ of p(N) takes the form $\mathbb{P} [N\leq p^{-1}(\cdot)]$ or $\mathbb{P} [N>p^{-1}(\cdot)]$ . Hence, for any $x\in{\mathbb{R}}$ , the density $q_{p(N)}$ of p(N) satisfies

\[ q_{p(N)}(x)=q_{N} (p^{-1}(x)) | (p^{-1})'(x)| =\dfrac{q_N (p^{-1}(x))}{|p'(p^{-1}(x))|}\leq \dfrac{1}{c_p\sigma\sqrt{2\pi}},\]

as the density of N, $q_N$ , is bounded above by $(\sigma\sqrt{2\pi})^{-1}$ .

5.4. CFs and distributions of near-normal random variables

Proposition 5.8. Let N be a normal random variable with mean $\mu$ and variance $\sigma^2$ , and let X be a continuous random variable. Let $\varphi_X$ , $\varphi_N$ and $\Phi_X$ , $\Phi_N$ denote the CFs and the distributions of X and N, respectively. Assume there exist constants $r>0$ , $\gamma\in (0,1)$ and a function $R\colon{\mathbb{R}}\,{\to}\,{\mathbb{R}}$ such that $ |\log\varphi_X(t)-\log\varphi_N(t)|\leq R(t)\leq \gamma\sigma^2 t^2/2$ holds on $|t|\leq r$ . Then

\[\sup_{x\in{\mathbb{R}}} |\Phi_{N}(x)-\Phi_{X}(x)|\leq \int_{-r}^{r}\dfrac{R(t)}{\pi |t|}\exp\bigg({-}\dfrac{(1-\gamma)\sigma^2t^2}{2}\bigg) \,{\mathrm{d}} t+\dfrac{12\sqrt{2}}{ \pi^{3/2}\sigma r}.\]

Proof of Proposition 5.8. The smoothing theorem implies

\[\sup_{x\in\mathbb{R}} |\Phi_{N}(x)-\Phi_{X}(x)| \leq\int_{-r}^{r}\dfrac{|\varphi_{N}(t)-\varphi_X(t)|}{\pi|t|}\,\mathrm{d}t+\dfrac{24}{\pi r}\sup_{x\in\mathbb{R}}|\Phi_N'(x)|. \]

Note that, for any $z\in\mathbb{C}$ , we have $|{\mathrm{e}}^z-1|\leq |z|\,{\mathrm{e}}^{|z|}$ . For $z\,:\!=\log(\varphi_X(t)/\varphi_N(t))$ , this implies

\[|\varphi_X(t)-\varphi_N(t)|\leq |\varphi_N(t)|\, |\log \varphi_X(t)-\log \varphi_N(t)| \exp(|\log \varphi_X(t)-\log \varphi_N(t)|) \quad \text{{for all} } t\in{\mathbb{R}}. \]

The result follows from this inequality, $\sup_{x\in{\mathbb{R}}}|\Phi_N'(x)|=1/(\sigma \sqrt{2\pi})$ , and $|\varphi_N(t)|={\mathrm{e}}^{-\sigma^2t^2/2}$ :

\begin{equation*} \int_{-r}^{r}{\frac{ |\varphi_{N}(t)-\varphi_{X}(t)|}{\pi |t|} \,{\mathrm{d}} t}\leq \int_{-r}^{r}{|\varphi_{N}(t)|\frac{R(t)}{\pi |t|}\,{\mathrm{e}}^{R(t)} \,{\mathrm{d}} t} \leq \int_{-r}^{r}\frac{R(t)}{\pi |t|}\exp\bigg({-}\frac{(1-\gamma)\sigma^2t^2}{2}\bigg) \,{\mathrm{d}} t.\nonumber \end{equation*}

Lemma 5.2. Let X be a random variable with finite mean $\mu$ , variance $\sigma^2$ , and absolute third central moment $\kappa\,:\!=\mathbb{E} [ |X-\mu|^3]$ . Then the characteristic function $\varphi_X$ of X satisfies

\[ \bigg|\log\varphi_X(t)- \bigg({\mathrm{i}}\mu t-\dfrac{\sigma^2}{2}t^2\bigg)\bigg|\leq \dfrac{\kappa|t|^3}{6}+\dfrac{\sigma^4t^4}{4}\quad\text{{for all} } t\in\bigg[-\dfrac{1}{\sigma},\dfrac{1}{\sigma}\bigg].\]

Proof. The result can be established by combining the elementary bound

\[\bigg|\mathbb{E}\bigg[{\mathrm{e}}^{{\mathrm{i}}(X-\mu)t}-\sum_{k=0}^n\dfrac{({\mathrm{i}} t)^n}{n!}(X-\mu)^n\bigg]\bigg|\leq \dfrac{|t|^{n+1}}{(n+1)!}\mathbb{E} [ |X-\mu|^{n+1}]\quad\text{{for all} } t\in{\mathbb{R}}\]

and the fact that $z\in\mathbb{C}$ , $|z|\leq 1/2$ implies $ |(\log(1+z)-z|\leq |z|^2$ (see [Reference Williams32, p. 188] for both).

Lemma 5.3. Let $N=N(0,\sigma^2)$ and let $u,v\in{\mathbb{R}}$ . The random variable $uN+vN^2$ has a characteristic function that satisfies

\begin{align*} \bigg|\log\varphi_{uN+vN^2}(t)-\bigg({\mathrm{i}} v\sigma^2t-\dfrac{u^2\sigma^2}{2}t^2\bigg)\bigg|\leq 2v^2\sigma^4t^2+2u^2|v|\sigma^4|t|^3\\ \text{{for all} } t\in\bigg[-\dfrac{1}{4|v|\sigma^2},\dfrac{1}{4|v|\sigma^2}\bigg]. \end{align*}

Proof. The CF $\varphi_{uN+vN^2}$ can be explicitly computed using standard complex analysis:

\[\varphi_{uN+vN^2}(t)=\mathbb{E} [{\mathrm{e}}^{{\mathrm{i}}(uN+vN^2)t}]=\dfrac{1}{\sqrt{1-2{\mathrm{i}} v\sigma^2t}}\exp\bigg({-}\dfrac{u^2\sigma^2t^2}{2(1-2{\mathrm{i}} v\sigma^2t)}\bigg)\quad \text{{for all} } t\in{\mathbb{R}}.\]

The rest can then be shown using the elementary inequalities: $z\in\mathbb{C}$ , $|z|\leq 1/2$ implies $ |(\log(1+z)-z|\leq |z|^2$ and $ |1/(1-z)-1|\leq 2|z|$ .