3.1. Motivation

Assume that X is stationary with marginal distribution function $F_X(x)=\mathbb P(X_0\le x)$ , $x\in\mathbb{R}$ , covariance function $C(t)=\textrm{cov}(X_0, X_t)$ , $t\in T$ , and moreover

(3) \begin{equation} \textrm{cov}_X(t,u,v)\ge 0 \mbox{ or } \le 0\quad \text{for all } t\in T, \ u,v\in\mathbb{R}.\end{equation}

Examples of X with this property are all PA or NA random functions. Applying [Reference Lehmann21, Lemma 2], we have (the equality is originally attributed to Hoeffding (1940))

\begin{equation*} C_X(t)=\int_{\mathbb{R}^2} \textrm{cov}_X(t,u,v) \, {\textrm{d}} u\,{\textrm{d}} v.\end{equation*}

Then X is long range dependent if

(4) \begin{equation}\int_T |C_X(t)|\, {\textrm{d}} t=\int_T\int_{\mathbb{R}^2}|\textrm{cov}_X(t,u,v)|\, {\textrm{d}} u\,{\textrm{d}} v\,{\textrm{d}} t=+\infty,\end{equation}

which agrees with the classical definition.

However, in Definition 1 we integrate $|\textrm{cov}_X(t,u,v)|$ with respect to a finite measure $\mu\times\mu$ instead of Lebesgue measure ${\textrm{d}} u \, {\textrm{d}} v$ . First, for infinite variance the right-hand side of (4) is often infinite, regardless of any dependence structure. As such, the classical definition of long memory is irrelevant in the infinite variance case. Second, our definition will have a natural link with the asymptotic behaviour of volumes of excursions of X above levels u, v. Recall the functional central limit theorem (CLT) for normed volumes of excursion sets of X at level u proved in [Reference Meschenmoser and Shashkin26] (see also [Reference Spodarev41, Theorem 9] for a generalization of this result to fields without a finite second moment). Namely, for a large class of weakly dependent stationary random fields X on $\mathbb{R}^d$ , the function

\[\int_{\mathbb{R}^d}\textrm{cov}_X(t,u,v)\,{\textrm{d}} t, \quad u,v\in\mathbb{R}\]

is the covariance function of the centred Gaussian process which appears as a limit of

(5) \begin{equation}\dfrac{\nu_d (\{ t\in [0,n]^d\colon X_t>u \}) -n^d \bar{F}_{X}(u) }{n^{d/2}}, \quad u\in\mathbb{R},\ n\to\infty\end{equation}

in $\mathcal{D}(\mathbb{R})$ equipped with the $J_1$ Skorokhod topology. If, in particular, the random field is PA or NA, then by the continuous mapping theorem we have

(6) \begin{equation}\dfrac{\int_{\mathbb{R}} \nu_d (\{ t\in [0,n]^d\colon X_t>u \}) \mu({\textrm{d}} u) -n^d \int_{\mathbb{R}} \bar{F}_{X}(u)\mu({\textrm{d}} u) }{n^{d/2}}\stackrel{d}{\longrightarrow} N(0,\sigma^2_{\mu,X})\end{equation}

as $n\to\infty$ for any finite measure $\mu$ with $\sigma^2_{\mu,X} $ as in Definition 1. So X is s.r.d. if the asymptotic covariance $\sigma^2_{\mu,X} $ in the central limit theorem (6) is finite for any finite integration measure $\mu$ prescribing the choice of levels u. In contrast,

\begin{equation*} \sigma^2_{\mu,X}=+\infty\end{equation*}

for $\mu=\delta_{\{ u_0 \}}$ means that a different normalization is needed in (5) and a non-Gaussian limit may arise.

Let us point out one possible interpretation of Definition 1 in a financial context. Assume $X=\{ X_t, t\in \mathbb Z\}$ to be a time series representing the stock price for which an American option at price $u_0>0$ , $t\in [0,n]$ , $n\in\mathbb N$ is issued. The customer may buy a call at price $u_0$ whenever $X_t>u_0$ for some $t\in[0,n]$ . Relation (6) with $\mu=\delta_{\{ u_0 \}}$ is written as

\[\dfrac{ \nu_1 (\{ t\in [0,n]\colon X_t>u_0 \}) -n \bar{F}_{X}(u_0) }{\sqrt{n}}\stackrel{d}{\longrightarrow} N(0,\sigma^2_{\delta_{\{ u_0 \}},X}),n \to \infty .\]

Then the long range dependence in the sense of Definition 1 of the stock price X (i.e. $\sigma^2_{\delta_{\{ u_0 \}},X} =+\infty$ ) means that the amount of time within [0,n] at which the option may be exercised is not asymptotically normal for large time horizons n. In contrast, the short range dependence of stock X means asymptotic normality of this time span for any price $u_0$ for which the option was issued, provided that X satisfies the conditions of [Reference Meschenmoser and Shashkin26] or [Reference Spodarev41].

In terms of potential theory, the value $\sigma^2_{\mu,X}$ in Definition 1 is the energy of measure $\mu$ with symmetric kernel $K(u,v)=\int_T |\textrm{cov}_X(t,u,v)|\,{\textrm{d}} t$ ; see [Reference Landkof19, page 77 ff.].

Self-similar random fields. We conclude this section with the formulation of long range dependence in a special case of self-similarity.

Let $X=\{X_t,t\in \mathbb{R}^d_+\}$ be a real-valued multi-self-similar random field. By definition, it is stochastically continuous and there exist numbers $H_1,\ldots,H_d>0$ such that, for a diagonal matrix $A=\mbox{diag}(a_1,\ldots,a_d)$ with $a_1,\ldots,a_d>0$ , we have

\[\{ X_{At}, \; t\in\mathbb{R}^d_+\} \stackrel{d}{=} \{ a_1^{H_1}\ldots a_d^{H_d} X_t, \; t\in\mathbb{R}^d_+\} .\]

Introduce the notation ${\bf 1}=(1,\ldots, 1)\in\mathbb{R}^d_+$ and ${\textrm{e}}^s=({\textrm{e}}^{s_1}, \ldots, {\textrm{e}}^{s_d} )$ for $s=(s_1,\ldots, s_d)\in\mathbb{R}^d$ . By [Reference Davydov and Paulauskas11, Proposition 6], the field

\[Y=\{ Y_s= {\textrm{e}}^{-\sum_{j=1}^d s_j H_j } X_{{\textrm{e}}^s}, \; s\in\mathbb{R}^d\}\]

is stationary. Using Definition 1 for Y together with the substitution $t_i={\textrm{e}}^{s_i}$ , $i=1,\ldots, d$ , we say that X is s.r.d. if, for any finite measure $\mu$ on $\mathbb{R}$ ,

\[\int_{\mathbb{R}^d_+}\int_{\mathbb{R}^2} \biggl| {\textrm{cov}}\biggl({\bf 1} (X_{\bf 1}>u),{\bf 1}\biggl(X_{\bf t}>v\cdot \prod_{j=1}^d t_j^{H_j}\biggr) \biggr) \biggr|\, \dfrac{\mu({\textrm{d}} u)\, \mu({\textrm{d}} v)\, {\textrm{d}} t}{\prod_{j=1}^d t_j } <+\infty,\]

where ${\textrm{d}} t={\textrm{d}} t_1\ldots {\textrm{d}} t_d$ means integration with respect to Lebesgue measure in $\mathbb{R}^d_+$ . Conversely, X is l.r.d. if the above integral is infinite for some finite measure $\mu$ on $\mathbb{R}$ .

3.2. Checking the short or long range dependence

Let $\mathbb P_{\mu}(\cdot)=\mu(\cdot)/\mu(\mathbb{R})$ denote the probability measure associated with the finite measure $\mu$ on $\mathbb{R}$ . Let U,V be two independent random variables with distribution $\mathbb P_\mu$ . Then the variance $\sigma^2_{\mu,X}$ from Definition 1 becomes

(7) \begin{equation}\dfrac{\sigma^2_{\mu,X}}{\mu^2(\mathbb{R})}= \int_T \mathbb E |{\textrm{cov}}_X(t,U,V)|\,{\textrm{d}} t= \int_T \mathbb E |F_{X_0,X_t}(U,V)-F_{X_0}(U)F_{X_t}(V)|\,{\textrm{d}} t.\end{equation}

This relation is useful for checking the short range dependence of X by showing the finiteness of $\sigma^2_{\mu,X}$ for any i.i.d. random variables U and V. Definition 1 is equivalent to the following lemma.

Lemma 1. A stationary real-valued random field X with marginal distribution function $F_X$ is s.r.d. in the sense of Definition 1 if

\[\int_T\int_{{(\operatorname{Im} F_X)^2}}| C_{0,t}(x,y)-xy |\, \mathbb P_0({\textrm{d}} x)\, \mathbb P_0({\textrm{d}} y)\,{\textrm{d}} t<+\infty\]

for any probability measure $\mathbb P_0$ on ${\operatorname{Im} F_X}$ , where $C_{0,t}$ is the copula of the bivariate distribution of $(X_0, X_t)$ , $t\in T$ , and $\operatorname{Im} F_X=F_X(\bar{\mathbb{R}})\subseteq [0,1]$ is the range of $F_X$ on $\bar{\mathbb{R}}=\mathbb{R}\cup\{+\infty\}\cup\{-\infty\}$ ; X is l.r.d. in the sense of Definition 1 if there exists a probability measure $\mathbb P_0$ on $\operatorname{Im} F_X$ such that the above integral is infinite.

Proof. By relation (7) and Sklar’s theorem (see e.g. [Reference Durante and Sempi14, Theorem 2.2.1]) we have, for any definite measure $\mu$ on $\mathbb{R}$ ,

\[\sigma^2_{\mu,X}={\mu^2(\mathbb{R})} \int_T\int_{\mathbb{R}^2}|C_{0,t}(F_X(u),F_X(v))-F_X(u)F_X(v)|\, \mathbb P_{\mu}({\textrm{d}} u)\, \mathbb P_{\mu}({\textrm{d}} v) \,{\textrm{d}} t.\]

The choice of $C_{0,t}$ is unique on $\operatorname{Im} F_X$ ; see [Reference Durante and Sempi14, Lemma 2.2.9]. Applying the substitution $x=F_X(u)$ , $y=F_X(v)$ , we get

\[\sigma^2_{\mu,X}={\mu^2(\mathbb{R})} \int_T\int_{(\operatorname{Im} F_X)^2}| C_{0,t}(x,y)-xy |\, \mathbb P_0({\textrm{d}} x)\, \mathbb P_0({\textrm{d}} y)\,{\textrm{d}} t,\]

where the probability measure $ \mathbb P_0$ has a cumulative distribution function $\mu ( (-\infty, F_X^-(x)) ) $ , $x\in[0,1]$ , and $F_X^-$ is the generalized inverse for $F_X$ .

Lemma 1 implies that the new definition of memory is marginal-free, i.e. independent of the distribution of marginals $F_X$ if $\operatorname{Im} F_X=[0,1]$ , which is the case for absolutely continuous $F_X$ . It essentially involves only the bivariate dependence structure encoded in the copula $C_{0,t}$ .

If condition (3) holds, then application of the Fubini–Tonelli theorem leads to

\begin{equation*}\sigma^2_{\mu,X}=\mu^2(\mathbb{R}) \int_T \textrm{cov} ( F_\mu (X_0), F_\mu (X_t) ) \, {\textrm{d}} t,\end{equation*}

where $F_\mu (x)=\mathbb P_\mu((-\infty, x))$ is the (left-side continuous) distribution function of probability measure $\mathbb P_\mu$ . In this case the s.r.d. condition $\sigma^2_{\mu,X}<+\infty$ reads as a classical covariance summability property of the subordinated random field $Y_t=F_\mu (X_t)$ , $t\in T$ .

By stationarity of X, we have $\textrm{cov}_X(t,u,v)=\textrm{cov}_X(-t,u,v)$ for any $t,-t\in T$ , $u,v\in\mathbb{R}$ . Hence, in order to show long range dependence for $T=\mathbb{R}$ , it is enough to check that

\[\int_0^\infty |\textrm{cov}_X(t,u_0,u_0)|\,{\textrm{d}} t=+\infty \]

for some $u_0\in\mathbb{R}$ . For $T=\mathbb Z$ it is sufficient to consider $\sum_{t=1}^\infty |\textrm{cov}_X(t,u_0,u_0)| =+\infty. $

3.2.1. Short range dependence for mixing random fields

Let $\mathcal{U}, \mathcal{V}$ be two sub- $\sigma$ -algebras of $\mathcal{F}$ . Introduce the z-mixing coefficient $z(\mathcal{U},\mathcal{V})$ (where $z\in\{\alpha,\beta,\phi, \psi, \rho \}$ ) as in [Reference Doukhan13, page 3]. For instance, it is given for $z=\alpha$ by

\[\alpha(\mathcal{U},\mathcal{V}) = \sup\{ | \mathbb P(U \cap V)-\mathbb P(U)\mathbb P(V) |\colon U\in \mathcal{U}, \;V\in \mathcal{V} \}.\]

Let $X=\{X_t, t\in T\}$ be a random field. Let $X_C=\{X_t, t\in C\}$ , $C\subset T$ , and let $\sigma_{X_\mathcal{C}}$ be the $\sigma$ -algebra generated by $X_C$ . If $|C|$ is the cardinality of a finite set C, then the z-mixing coefficient of X is given by

\[ z_X(k, u, v) = \sup\{z(\sigma_{X_A},\sigma_{X_B})\colon d(A,B) \geq k,\; |A| \leq u, |B| \leq v\},\]

where $ u,v \in \mathbb N $ and d(A,B) is the Hausdorff distance between finite subsets A and B generated by the metric on $\mathbb{R}^d$ . The interrelations between different mixing coefficients $z_X$ , $z\in\{\alpha,\beta,\phi, \psi, \rho \}$ are given in [Reference Doukhan13, Proposition 1], for example.

We state the result that links mixing properties and the short range dependence. The field X may be non-Gaussian and have infinite variance.

Theorem 1. Let $X = \{X_t, t\in T\}$ be a stationary random field with z-mixing rate satisfying $\int_T z_X(\|t\|,1,1) \, {\textrm{d}} t < +\infty$ , where $z\in\{\alpha,\beta,\phi, \psi, \rho \}$ . Then X is s.r.d. in the sense of Definition 1 with

\[\int_T\int_{\mathbb{R}^2} |\textrm{cov}_X(t, u, v)| \, {\mu}({\textrm{d}} u) \, {\mu}({\textrm{d}} v) \, {\textrm{d}} t \leq 8\int_T z_X(\|t\|,1,1) \, {\textrm{d}} t \cdot \mu^2(\mathbb{R}) < +\infty.\]

Proof. Without loss of generality, we prove the result for $\alpha$ -mixing X. Introduce random variables $\xi(u)={\bf 1}(X_0>u)$ , $\eta(v)={\bf 1}(X_t>v)$ , where $t \in T$ , $u,v \in \mathbb{R}.$ Then, by the covariance inequality in [Reference Doukhan13, Theorem 3] connecting the covariance of random variables with their mixing rates, we have

\begin{align*} &\int_T\int_{\mathbb{R}^2} |\textrm{cov}_X(t, u, v)| \mu(\textrm{d} u) \mu( \textrm{d} v) \,\textrm{d} t = \int_T\int_{\mathbb{R}^2} |\textrm{cov}(\xi(u), \eta(v))| \mu(\textrm{d} u) \mu( \textrm{d} v) \,\textrm{d} t \\[3pt] &\quad \le 8 \int_T \alpha (\sigma_{X_0},\sigma_{X_t})\,\textrm{d} t \int_{\mathbb{R}^2 } \| \xi(u) \|_{\infty} \| \eta(v)\|_{\infty} \mu ({\textrm{d}} u) \mu ({\textrm{d}} v) \\[3pt] &\quad \leq 8\int_T \alpha_X(\|t\|,1,1)\,\textrm{d} t \cdot \mu^2(\mathbb{R}) < +\infty,\end{align*}

where $ \| Y \|_{\infty} = \mbox{Ess-sup} (Y)$ .

To illustrate the above theorem, we let $Y=\{ Y_t, \, t\in \mathbb N \}$ denote a stationary almost surely (a.s.) non-negative $\psi$ -mixing random sequence with univariate cumulative distribution function $F_Y$ and $\int_{\mathbb{R}^d} \psi_Y(\|t\|,1,1) \, {\textrm{d}} t < +\infty.$ Examples of $\psi$ -mixing random sequences can be found for example in [Reference Doukhan13, Example 4] (see also references therein), [Reference Heinrich16, Theorem 2.2], [Reference Rapaport29, Proof of Claim 2.5], [Reference Bradley6], and [Reference Samur38, pages 54–55]. Let $F^{-1}_Z$ be the quantile function of a random variable Z with $\mathbb E Z^2=+\infty$ . Set $G(x)=F^{-1}_Z(F_Y(x))$ , $x\ge 0$ ; then $X_t=G(Y_t)$ , $t\in \mathbb N$ is $\psi$ -mixing as well. Moreover, it is s.r.d. by the last theorem and has infinite variance because of $X_0\stackrel{d}{=} Z$ .

Remark 1. For a Gaussian $\phi$ -mixing random field X, the statement of Theorem 1 is trivial, since such an X is m-dependent [Reference Ibragimov and Linnik17, Theorem 17.3.2], and the integral $\int_0^\infty |\textrm{cov}_X(t,u,v)|\,{\textrm{d}} t$ in Definition 1 is bounded by 2m for any $u,v\in\mathbb{R}$ .

3.3. Subordinated Gaussian random fields

Recall that $\varphi(x) $ is the density of the standard normal law. We use the notation $\Phi(x)$ for its cumulative distribution function. Introduce the Hermite polynomials $H_n$ of degree $n\in \mathbb N_0$ by

\[H_n(x)=(-1)^n \varphi^{(n)}(x) / \varphi (x),\]

where $\varphi^{(n)}$ is the nth derivative of $\varphi$ . Clearly

\[H_0(x)=1,\quad H_1(x)=x,\quad\ H_2(x)=x^2-1,\quad H_3(x)=x^3-3x, \quad \ldots . \]

For even orders n, Hermite polynomials are even functions, whereas for odd n they are odd functions. It is well known that Hermite polynomials form an orthogonal basis in $L^2_{\varphi}(\mathbb{R})$ . For any function $f\in L^2_{\varphi}(\mathbb{R})$ with $\langle f, 1\rangle_{\varphi} = 0$ , let

\[\operatorname{rank} (f)=\min\{n\in\mathbb N\colon \langle f, H_n\rangle_{\varphi} \neq 0 \}\]

be the Hermite rank of f. Furthermore, the Hermite rank can also be defined for functions $f\not\in L^2_{\varphi}(\mathbb{R})$ , as long as $\langle |f|^{1+\theta}, \varphi\rangle<\infty$ for some $\theta\in (0,1)$ ; see [Reference Sly and Heyde40] or [Reference Beran, Feng, Ghosh and Kulik5, Section 4.3.5].

Let $Y = \{Y_t, t\in T\}$ be a stationary centred Gaussian real-valued random field with $\textrm{var}\, Y_t=1$ and $C_Y(t)=\textrm{cov}(Y_0,Y_t)$ , $t\in T$ . The subordinated Gaussian random field X is defined by $X_t=G(Y_t)$ , $ t\in T$ , where $G\colon \mathbb{R}\rightarrow \operatorname{Im}(G)\subseteq\mathbb{R}$ is a measurable function.

Assume first that X is square-integrable. The following lemma is proved in [Reference Rozanov33, Lemma 10.2].

Lemma 2. Let $Z_1, Z_2$ be standard normal random variables with $\rho = \textrm{cov}(Z_1,Z_2)$ , and let F, G be functions satisfying $\mathbb E F^2(Z_1), \mathbb E G^2(Z_1) < +\infty$ . Then

\[\textrm{cov}(F(Z_1), G(Z_2)) = \sum_{k=1}^{\infty}\dfrac{\langle F, H_k\rangle_{\varphi} \langle G, H_k\rangle_{\varphi} }{k!}\rho^k . \]

Let $C_X(t)=\textrm{cov}(X_0,X_t)$ , $t\in T$ . Assuming $C_Y(t)\ge 0$ for all $t\in T$ and applying this lemma to our subordinated process $X=G(Y)$ we get that it is s.r.d. in the sense of Definition 1 if

(8) \begin{equation} \int_T|C_X(t)| \, {\textrm{d}} t = \sum_{k=1}^{\infty}\dfrac{\langle G, H_k\rangle_{\varphi}^2}{k!}\int_T C_Y^k(t)\, {\textrm{d}} t <+\infty.\end{equation}

We shall see that an analogous result holds also if X has no finite second moment. Introduce the condition

( $\boldsymbol{\rho}$ ) $ |C_Y(t)|<1$ for all $t\neq 0$ if T is countable, and for $\nu_d$ -almost every $t\in T$ if T is uncountable.

The following result gives the conditions for short range dependence of a subordinated Gaussian random field, without a moment assumption. Its proof is given in the Appendix.

Theorem 2. Let Y be the Gaussian random field introduced above. Let X be a subordinated Gaussian random field defined by $X_t=G(Y_t)$ , $t\in T$ , where G is a right-continuous strictly monotone (increasing or decreasing) function. Assume that condition ( $\boldsymbol{\rho}$ ) holds. Let

(9) \begin{equation}b_k(\mu)=\biggl(\int_{\operatorname{Im}(G)}H_{k}(G^-(u))\varphi(G^-(u)) \, \mu({\textrm{d}} u)\biggr)^2,\end{equation}

where $G^-$ is the generalized inverse of G if G is increasing or the generalized inverse of $-G$ if G is decreasing. Then X is s.r.d. in the sense of Definition 1 if and only if

(10) \begin{equation}\sum_{k=1}^{\infty}\dfrac{b_{k-1}(\mu)}{k!} \int_T |C_Y(t)| C_Y^{k-1}(t) \, {\textrm{d}} t <+\infty\end{equation}

for any finite measure $\mu$ on $\mathbb{R}$ .

Let us illustrate the last point of Remark 2 with an example.

Example 1. Let $G(x)={\textrm{e}}^{x^2/(2\alpha)}$ , $\alpha>0$ , $T=\mathbb{R}^d$ . Then it is easy to see that

\[\mathbb P (|X_0|>x)=L(x) x^{-\alpha},\]

where $L(x)=\sqrt{ 2/(\pi \log x) } $ . For $\alpha\in(1,2]$ , $\mathbb E \, X_0<\infty$ , $\mathbb E \, X_0^2=+\infty$ .

To compute $b_{2k-1}(\mu)$ , we notice that

\[\sqrt{b_{2k-1}(\mu)}=\dfrac{1}{\sqrt{2\pi}}\int_1^\infty u^{-\alpha} H_{2k-1} (\sqrt{2\alpha \log u}) \, \mu({\textrm{d}} u), \quad k\in \mathbb N.\]

Using the upper bound $ | H_{2k-1}(x) | \le x \,{\textrm{e}}^{x^2/4} (2k-1)!! /4$ , $x\ge 0$ from [Reference Abramowitz and Stegun1, page 787], one can show that

\begin{align*}b_{2k-1}(\mu) &\le \dfrac{ \alpha}{16\pi} [(2k-1)!!]^2 \biggl( \int_1^\infty u^{-\alpha/2} \sqrt{\log u} \, \mu({\textrm{d}} u) \biggr)^2 \\[3pt] &\le \dfrac{ \alpha }{ 4\pi } \mu^2 ([1,+\infty)) [ (2k-1)!!]^2 \\[3pt] &<+\infty\end{align*}

for all $ k\in\mathbb N.$

We note that the use of the finite measure $\mu$ is crucial here, since for the Lebesgue measure the integral $\int_1^\infty u^{-\alpha/2} \sqrt{\log u} \, \mu({\textrm{d}} u)$ is infinite for $\alpha\leq2$ .

Now, by Stirling’s formula [Reference Andrews, Askey and Roy4, Theorem 1.4.2], we get

\begin{equation*} \dfrac{[(2k-1)!!]^2 }{ (2k)! } \sim \dfrac{c_3}{\sqrt{k}}, \quad k\to +\infty\end{equation*}

for $c_3>0$ , so

(12) \begin{equation}\dfrac{b_{2k-1}(\mu)}{(2k)!}= O( \dfrac{1}{\sqrt{k} }) , \quad k\to +\infty.\end{equation}

Assume that $C_Y(t)\sim \|t\|^{-\eta}$ as $\|t\|\to +\infty$ , $\eta>0$ . Then $X={\textrm{e}}^{Y^2/(2\alpha)}$ , $\alpha >0$ , is

• • l.r.d. if $\eta\in (0, d/2]$ , since then

  \begin{equation*}\sum_{k=1}^{\infty}\dfrac{b_{2k-1}(\mu)}{(2k)!} \int_T C_Y^{2k}(t) \, {\textrm{d}} t =+\infty;\end{equation*}

• • s.r.d. if $\eta> d/2$ , since then we have

  \[\int_{\mathbb{R}^d} C_Y^{2k}(t)\, {\textrm{d}} t =O(k^{-1}) \quad \text{as } k\to +\infty,\]
  and the series (11) behaves like ${\sum_{k=1}^\infty k^{-3/2} <+\infty.}$

Here the source of long memory of X is the l.r.d. field Y. If $\alpha>2$ the variance of $X_0$ is finite, and our results agree with the definition in (1) by relation (8) if we note that $\operatorname{rank} (G)=2$ . However, the main point of this example is that we have the same transition from short to long memory (i.e. $\eta=d/2$ ) for both finite and infinite variance fields.

Note that for $\eta\in (d/2, d)$ the Gaussian field Y is l.r.d. but the subordinated field $X={\textrm{e}}^{Y^2/(2\alpha)}$ is s.r.d. This agrees with the classical theory for finite variance, but is novel for infinite variance.

3.4. Stochastic volatility models

We present a way of constructing random fields with long memory by introducing a random volatility $G(Y_t)$ (being a deterministic function of a random scaling field $Y=\{ Y_t, t\in T \}$ ) of a random field $Z=\{ Z_t, t\in T \}$ . We assume that Y and Z are independent. An overview of random volatility models and their applications in finance can be found in [Reference Shephard39] and [Reference Andersen, Davis and Mikosch3, Part II], for example. For each $t\in T$ , $X_t=G(Y_t)Z_t$ is a scale mixture of $G(Y_t)$ and $Z_t$ ; see [Reference Steutel and van Harn42, Chapter VI, page 345]. Let $\bar{F}_Z=1-F_Z$ be the marginal tail distribution function of $Z_t$ for stationary Z.

For a finite measure $\mu$ , introduce the functional

\[{D_{\mu}} (G(Y),Z_0) =\int_T\int_{\mathbb{R}^2}\textrm{cov} ( \bar{F}_Z (u/G(Y_0)),\bar{F}_Z (v/G(Y_t)) ) \mu(\textrm{d} u) \mu( \textrm{d} v)\,{\textrm{d}} t.\]

The next lemma follows trivially from relation (2), independence of Y and Z, and Tonelli’s theorem.

Lemma 3. Let a random field $X=\{ X_t, t\in T \}$ be given by $X_t=G(Y_t)Z_t$ , where $Y=\{ Y_t, t\in T \}$ and $Z=\{ Z_t, t\in T \}$ are independent stationary random fields, Z has property (3), $G\colon \mathbb{R}\to\mathbb{R}_{\pm}$ and $\mathbb P (G(Y_t)=0)=0$ for all $t\in T$ . Then

(13) \begin{align}&\int_T \int_{\mathbb{R}^2}\textrm{cov}_X(t,u,v)\, \mu({\textrm{d}} u)\,\mu({\textrm{d}} v)\,{\textrm{d}} t \notag\\[3pt] &\quad = D_{\mu} (G(Y),Z_0) + \int_T \int_{\mathbb{R}^2}\mathbb E [ \textrm{cov}_Z(t,u/G(Y_0),v/G(Y_t))]\, \mu({\textrm{d}} u)\,\mu({\textrm{d}} v)\,{\textrm{d}} t.\end{align}

Let us illustrate the use of Lemma 3.

The above corollary evidently holds true if, for example, $Z_0\sim \textrm{Exp}(\lambda)$ , $$A \sim $$ Fréchet(1) for any $\lambda>0$ . It also clearly applies to a sub-Gaussian random field X, where $A=\sqrt{B}$ , $B\sim S_{\alpha/2} ( ( \cos {{\pi\alpha}/{\pi\alpha}}) ^{2/\alpha},1,0 ) $ , $\alpha \in (0,2)$ , and Z is a centred stationary Gaussian random field with covariance function $C(t)\ge 0$ for all $t\in T$ and a non-degenerate tail $\bar{F}_Z$ .

The following corollary describes the situation where a light-tailed Y is responsible for the long range dependence of X while Z is responsible for heavy tails.

Now we scale a l.r.d. (possibly heavy-tailed) random field Z by a random volatility G(Y) being a subordinated Gaussian random field.

Lemma 4. Let $X_t=G(Y_t) Z_t$ be a random field as in Lemma 3. Assume further that Y is a centred Gaussian random field with unit variance and covariance function $\rho(t)\ge 0$ satisfying condition ( $\boldsymbol{\rho}$ ). Then

\begin{equation*}D_{\mu} (G(Y),Z_0) = \sum_{k=1}^\infty \dfrac{ \bigl( \int_{\mathbb{R}} \langle \bar{F}_Z (u/G(\cdot)), H_k(\cdot) \rangle_\varphi \, \mu({\textrm{d}} u) \bigr)^2 }{k!}\int_T \rho^k(t) \, {\textrm{d}} t .\end{equation*}

The following example illustrates our definition of long range dependence in the context of a popular long memory stochastic volatility model that is used in econometrics to model log-returns of stocks; see [Reference Beran, Feng, Ghosh and Kulik5, page 70 ff.] and references therein.

Example 2. Assume that $X=\{X_t,t\in\mathbb Z\}$ has a form $X_t={\textrm{e}}^{Y_t^2 /4}Z_t$ , where $Z_t$ is a sequence of i.i.d. random variables with finite moment of order $2+\delta$ for some $\delta>0$ , while $Y_t$ is a centred stationary Gaussian PA long memory sequence with unit variance and covariance function $C_Y$ satisfying condition ( $\boldsymbol{\rho}$ ). Both sequences $Z_t$ and $Y_t$ are assumed to be independent of each other. From Example 1 we know that ${\textrm{e}}^{Y_0^2 /4}$ is regularly varying with index $\alpha=2$ . By Breiman’s lemma the tail distribution function of $|X_0|$ is also regularly varying with index $\alpha =2$ , and hence $X_0$ has infinite variance. Choose $\mu=\delta_{\{ u_0\}} $ for some $u_0\in\mathbb{R}$ . Lemmas 3 and 4 yield

(14) \begin{equation}\sum_{t=1}^{\infty} \int_{\mathbb{R}^2} {\textrm{cov}}_X(t,u,v) \mu({\textrm{d}} u)\mu({\textrm{d}} v) = \sum_{k=1}^\infty \dfrac{ \langle \bar{F}_Z(u_0/G), H_k\rangle_\varphi^2 }{k!} \sum_{t=1}^{\infty} C_Y^k(t) ,\end{equation}

where $G(x)={\textrm{e}}^{x^2/4}$ . Since $\bar{F}_Z(u_0/G)$ is symmetric, monotone non-decreasing, and bounded, we get $\langle \bar{F}_Z(u_0/G), H_{k}\rangle_\varphi=0$ for all odd k, and it is finite for all even $k\in\mathbb N$ . Moreover, by Lemma 5(ii) we have $\operatorname{rank}(\bar{F}_Z(u_0/G) )=2$ . It is clear that X is l.r.d. if $\sum_{t=1}^\infty\rho^2(t)=+\infty$ . In particular, if $C_Y(t)\sim |t|^{-\eta}$ as $|t|\to\infty$ , then long range dependence occurs if $\eta\in (0,1/2]$ . Again, similarly to Example 1, the point here is that we obtain long memory for both finite and infinite variance.

4.1. Limit theorems for subordinated Gaussian processes

Let $X=\{ X_t, t\in\mathbb{R}^d \}$ , where $X_t= G(Y_t)$ and $Y=\{ Y_t, t\in \mathbb{R}^d \}$ is a stationary isotropic l.r.d. centred Gaussian random field with covariance function $C_Y(t)= \|t\|^{-\eta}L(\|t\|)$ , $\eta\in (0,d/q)$ (see [Reference Ivanov and Leonenko18], [Reference Leonenko22], and [Reference Leonenko and Olenko23]). Here $\mathbb E G^2(Y_0)<+\infty$ and q is the Hermite rank of G. Under some technical assumptions on the spectral density $f(\lambda)$ of Y (see [Reference Leonenko and Olenko23, Assumption 2]),

\begin{align*} n^{q\eta/2-d}L^{-q/2}(n)\int_{W_n}G(Y_t)\, {\textrm{d}} t \stackrel{d}{\longrightarrow} R, \quad n\to+\infty,\end{align*}

where

(16) \begin{align} R &= ( \gamma(d,\eta) ) ^{q/2}\int^\prime_{\mathbb{R}^{dq}} \int_W \,{\textrm{e}}^{{\textrm{i}}\langle \lambda_1+\ldots+\lambda_q,u\rangle} \,{\textrm{d}} u \dfrac{\tilde{B}(d\lambda_1) \ldots \tilde{B} (d\lambda_q)}{ ( \| \lambda_1\| \cdot \ldots \cdot \| \lambda_q\| ) ^{(d-\eta)/2}},\nonumber\\[3pt] \gamma(d,\eta) &=\dfrac{\Gamma ( (d-\eta)/2 ) }{2^\eta \pi^{d/2} \Gamma(\eta/2)},\end{align}

and $\int^\prime_{\mathbb{R}^{dq}} $ is the multiple Wiener–ItÔ integral with respect to a complex Gaussian white noise measure $\tilde{B}$ (with structural measure being the spectral measure of Y; see [Reference Ivanov and Leonenko18, Section 2.9]). It is easy to see that for $q=1$ the distribution of R is Gaussian. However, the normalization $n^{\eta/2-d} L^{-1/2}(n)$ differs from the usual CLT normalizing factor $n^{-d/2}$ , which agrees with the fact that X is l.r.d. in the sense of the usual definition as in (1). For $q\ge 2$ , we get a q-Rosenblatt-type distribution for R; see [Reference Veillette and Taqqu43] and [Reference Leonenko, Ruiz-Medina and Taqqu24] and references therein for its properties in the case $q=2$ .

4.1.1 Volume of level sets

We apply the above observations to level sets. Assume $G\colon \mathbb{R}\to\mathbb{R}$ to be a monotone right-continuous function such that $\mathbb E |G(Y)|^{1+\theta}<+\infty $ with $\theta\in(0,1)$ . Let the variance of $X_0$ be infinite. For any $u\in\mathbb{R}$ introduce the function $g_u(x)=\zeta_{G,1,u} (x)$ , where $\zeta_{G,1,u} $ is given in (15). By Remark 3(i), the Hermite ranks of G and $g_u$ are equal to one. If $\eta\in (0,d)$ then

\begin{align*}\dfrac{\int_{W_n}g_u(Y_t)\, {\textrm{d}} t}{n^{d-\eta/2}L^{1/2}(n)} = \dfrac{ \int_{W_n} {\bf 1} (G(Y_t)>u) \, {\textrm{d}} t - \nu_d(W_n) \mathbb P (G(Y_0)>u)}{n^{d-\eta/2}L^{1/2}(n)} \stackrel{d}{\longrightarrow} R\end{align*}

as $n\to+\infty$ , where R is given in (16). The normalization in this limit theorem is not of CLT type $n^{-d/2}$ , which should be attributed to the l.r.d. case. Let us compare this behaviour with Definition 1. As an example, we consider

\[G(x)=\textrm{sign} (x) ( {\textrm{e}}^{x^2/\beta^2}-1) , \quad x\in\mathbb{R}\]

for some $\beta>\sqrt{2(1+\theta)}$ . Note that it is possible that the variance of $X=G(Y)$ is infinite. Set $\mu=\delta_{\{ 0 \} }$ . By Remark 2(i) we get $b_k(\mu)=H_k^2(0)/(2\pi)<+\infty$ for any $k\ge 0$ , $b_0>0$ , $b_1= 0$ , etc. By the choice $ C_Y(t)= \|t\|^{-\eta}L(\|t\|)$ , $\eta\in (0,d)$ we get that $\int_{\mathbb{R}^d} |C_Y(t)|\, {\textrm{d}} t = +\infty$ , and the series (10) diverges. Then X is l.r.d. in the sense of Definition 1 for $\eta\in (0,d)$ , which is in accordance with the above limit theorem.

4.1.2. Empirical mean: infinite variance case

In this section we show that Definition 1 cannot be linked to the behaviour of integrals or partial sums of the field X if X has infinite variance. For that, we use the framework of time series $X=\{ X_t, \ t\in\mathbb Z \}$ , where many more models have been widely explored, as compared to (continuous-time) random fields.

Consider (as in Section 3.3) a subordinated time series $X_t=G(|Y_t|)$ , $t\in\mathbb Z$ , where $\{Y_t, \: t\in\mathbb Z\}$ is a centred Gaussian long memory linear time series with non-decreasing covariance function $C_Y(t)=\textrm{cov}(Y_0,Y_t)\sim |t|^{-\eta}L(t) $ , $t\to+\infty$ , $\eta\in(0,1)$ , and such that $\mathbb P(|X_0|>x)\sim x^{-\alpha}L(x)$ , $\alpha\in (0,2)$ . It is further assumed that G has Hermite rank q. By Corollary 1(ii), X is short range dependent in the sense of Definition 1 whenever, for any finite measure $\mu$ on $\mathbb{R}$ ,

\begin{equation*} \sum_{k=1}^{\infty}\dfrac{b_{2k-1}(\mu)}{(2k)!} \sum_{t=1}^{\infty} C_Y^{2k}(t) <+\infty.\end{equation*}

We note that

(17) \begin{equation} \sum_{t=1}^{\infty} C_Y^{2k}(t) \le c_0 \int_{1}^{\infty} \dfrac{L^{2k}(t)}{ t^{2k\eta}}\, {\textrm{d}} t \le \int_{1}^{\infty} \dfrac{c_1 \, {\textrm{d}} t}{ t^{2k(\eta-\delta)}}, \quad k\in \mathbb N, \end{equation}

where $\delta>0$ is arbitrary and $c_0, c_1>0$ are some constants. The second inequality holds since $L(t)\le c_2 t^\delta$ for $t\ge t_0$ , where $t_0>0$ is large enough and $c_2=c_2(\delta,t_0)=(1+\delta) L(t_0)/t_0^\delta \le 1$ for large $t_0$ ; see [Reference Resnick30, Proposition 2.6]. The right-hand side of (17) is finite and equal to $O(1/k)$ whenever $\eta\in(1/2,1)$ , since $\delta>0$ can be chosen arbitrarily small. The series in (17) diverges if $\eta\in (0,1/2)$ . If $\eta=1/2$ , the summability of the series in (17) depends on the particular form of the slowly varying function L and will not be discussed here.

Thus, for $\eta\in(1/2,1)$ , X is s.r.d. whenever

(18) \begin{equation}\sum_{k=1}^{\infty}\dfrac{b_{2k-1}(\mu)}{(2k)! k} <+\infty \end{equation}

for any finite measure $\mu$ .

Now we have to consider a special example of function G in order to get more explicit results for the s.r.d. case. As in Example 1, set $G(x)={\textrm{e}}^{x^2/(2\alpha)}$ , $\alpha\in(0,2]$ . By relation (12), condition (18) is satisfied for $\eta\in(1/2,1)$ , hence X is s.r.d. in the sense of Definition 1 if $\eta\in(1/2,1)$ and l.r.d. if $\eta\in (0,1/2)$ .

Let us compare this result with the limiting behaviour of the partial sums $S_n=\sum_{t=1}^n (X_t-\mathbb E[X_t])$ as given in [Reference Sly and Heyde40] and [Reference Beran, Feng, Ghosh and Kulik5, Section 4.3.5]; see Table 1. There some discrepancies are seen, that is, Definition 1 does not agree with the asymptotic behaviour of $S_n$ .

Table 1.

Short or long memory of $X_t={\textrm{e}}^{Y_t^2/(2\alpha)}$ in the infinite variance case $\alpha\in(1,2)$ depending on the long memory parameter $\eta$ of Y according to [Reference Sly and Heyde40].

4.2. Limit theorems for the integrals of functionals of l.r.d. random volatility fields

In this section we will justify that our definition of long range dependence is in agreement with limit theorems for volumes of level sets for random volatility models. Unlike the subordinated Gaussian case, where the limiting results are known, a general asymptotic theory has to be developed.

Let X be a random volatility field of the form $X_t=G(Y_t)Z_t$ , $t\in \mathbb Z^d$ , where

• • $\{G(Y_t),t\in \mathbb{R}^d\}$ is a subordinated Gaussian measurable random field, which is sampled at points $t\in\mathbb Z^d$ ,

• • $\{Z_t,t\in \mathbb Z^d\}$ is white noise,

• • the random fields Y and Z are independent.

Our goal is to prove limit theorems for $\sum_{t\in W_n}g(X_t)$ as $n\to\infty$ , where $W_n=[-n,n]^d \cap \mathbb Z^d$ and g is a real-valued Borel-measurable function such that

(19) \begin{align}\mathbb E [g(X_0)]=0,\quad \mathbb E [g^2(X_0)]>0 .\end{align}

Introduce the function

\[\xi(y)= \mathbb E [g(G(y) Z_0)] .\]

It follows from (19) that for $\nu_1$ -almost every $y\in \mathbb{R}$

(20) \begin{align}\xi(y)<\infty.\end{align}

By (19) we also have $\mathbb E[\xi(Y_0)]=0$ . Let

\[J(m)=\langle \xi, H_m\rangle_\varphi = \mathbb E[H_m(Y_0) \, g(G(Y_0)Z_0)]\;\]

be the mth Hermite coefficient of $\xi$ . We recall that a sufficient condition for the finiteness of J(m) is

(21) \begin{align}\mathbb E [|g(X_0) |^{1+\theta}]= \mathbb E [|\xi(Y_0) |^{1+\theta}]=\mathbb E[| \mathbb E[g(G(Y_0)Z_0)\mid \mathcal{Y}]|^{1+\theta}]<\infty\end{align}

for some $\theta\in(0,1]$ , where $\mathcal{Y}$ is a sigma-field generated by the entire sequence Y. Let $\operatorname{rank} (\xi)= q$ . Furthermore, set

\[m(y,Z_t)=g(G(y)Z_t) - \mathbb E[g(G(y)Z_t)]=g(G(y)Z_t) - \xi(y),\]

which is almost everywhere finite by (20), and $\chi(y)=\mathbb E[m^2(y,Z_0)].$ We also assume

(22) \begin{align}\mathbb E[\chi^3(Y_0)]<\infty.\end{align}

Note that under (22), using the Lyapunov inequality on a space of finite measure and the stationarity of $Y_t$ , we have for any finite subset $I\subset \mathbb Z^d$

\begin{align*}\mathbb E\biggl[\biggl(\sum_{t\in I}\chi(Y_t)\biggr)^3\biggr]<\infty.\end{align*}

The following result shows that the limiting behaviour is primarily determined by the function $\xi$ , with $\xi\equiv 0$ being the boundary case.

Assume that (19), (21) with $\theta=1$ , and (22) hold.

1. (i) If $\xi(y)\equiv 0$ , then

   (23) \begin{align}n^{-d/2}\sum_{t\in W_n}g(X_t) \stackrel{d}{\longrightarrow} \mathcal{N}(0,\sigma^2), \quad n\to+\infty,\end{align}
   where $\sigma^2=\mathbb E[g^2(X_0)] 2^d>0$ .

2. (ii) If $\xi(y)\not\equiv 0$ , then

   (24) \begin{align}n^{q\eta/2-d}L^{-q/2}(n)\sum_{t\in W_n} g(X_t) \stackrel{d}{\longrightarrow} R, \quad n\to+\infty,\end{align}
   where the random variable R is given in (16) with $W=[-1,1]^d$ .

Example 5. Assume that $g(y)=g_u(y)={\bf 1} \{y>u\}-\mathbb P ( G(Y_0)Z_0>u) $ , where G is non-negative or non-positive $\nu_1$ -a.e. Then

\[\xi(y)= \mathbb E[{\bf 1} \{G(y)Z_0>u\} ]-\mathbb P ( G(Y_0)Z_0>u) \not\equiv 0\]

if $u\neq 0$ , so case (24) applies. If $u=0$ , then $\xi(y)\equiv 0$ (compare Remark 3(iii)), so case (23) holds true.

Example 6. Let the random volatility field $X_t=G(|Y_t|)Z_t$ , $t\in\mathbb Z^d$ be as in Lemma 4, where $\{ Z_t\}$ is heavy-tailed white noise, $\mathbb E Z_0^2=+\infty$ . Let Y satisfy the assumptions of Theorem 3. Choose $G(x)\ge 0$ as in Lemma 5(ii), and let $C_Y(t)\sim \|t\|^{-\eta}$ as $ \|t\|\to+\infty$ be non-negative. Similarly to Example 2, an analogue of relation (14) holds true: for $\mu=\delta_{\{u_0\}}$ , $u_0>0$ we have

\begin{equation*}\sum_{t\in \mathbb Z^d, \, t\neq 0}\int_{\mathbb{R}^2} {\textrm{cov}}_X(t,u,v) \mu({\textrm{d}} u)\mu({\textrm{d}} v) = \sum_{k=1}^\infty \dfrac{ \langle \bar{F}_Z(u_0/\widetilde{G}), H_k\rangle_\varphi^2 }{k!} \sum_{t\in \mathbb Z^d, \, t\neq 0}\ C_Y^k(t) ,\end{equation*}

where $\widetilde{G}(y)=G(|y|)$ , $y\in\mathbb{R}$ . Since $\operatorname{rank} (\bar{F}_Z(u_0/\widetilde{G}) )=2$ , X is l.r.d. in the sense of Definition 1 if $\sum_{t\in \mathbb Z^d, \, t\neq 0} C_Y^{2}(t) =+\infty$ , i.e. if $\eta<d/2$ .

Consider function $\xi $ from Example 5 with $u=u_0>0$ and $\widetilde{G}$ instead of G. By Lemma 5(ii), $\operatorname{rank} (\xi )=2$ . By Theorem 3 and Example 5, the asymptotic behaviour of the cardinality of the level sets of X at level $u_0$ is of l.r.d. type if $\eta\in(0,d/2)$ , which is in agreement with our definition.

Remark 4. We would like to connect the assumption $\xi\equiv 0$ to our definition. Let g,h be functions such that $\mathbb E[g(X_0)]=\mathbb E[h(X_0)]=0$ . If $\mathbb E [g(X_0)h(X_t)]<\infty$ for all t, and $\mathbb E[g(G(y)Z_0)]=\mathbb E[h(G(y)Z_0)]=0$ for all y, then for $t\not=0$

\begin{align*} \textrm{cov}(g(X_0),h(X_t))=\int\int\mathbb E [g(G(y_0)Z_0)] \mathbb E[h(G(y_t)Z_t)] \mathbb P_{Y_0,Y_t}({\textrm{d}} y_0,{\textrm{d}} y_t)=0 ,\end{align*}

where $\mathbb P_{Y_0,Y_t}$ is the joint law of $(Y_0,Y_t)$ . In particular, take

\[g(x)=g_u(x)={\bf 1}(x>u)-\mathbb P(X_0>u), \quad h(x)=h_v(x)={\bf 1}(x>v)-\mathbb P(X>v).\]

Then

\[\sigma_{\mu,X}^2=\sum_{t\in\mathbb Z^d, \, t\neq 0} \int_{\mathbb{R}^2} |\textrm{cov}(g_u(X_0),h_v(X_t))|\, {\textrm{d}} u \, {\textrm{d}} v=0,\]

and the random field X is s.r.d. according to Definition 1 for $\xi\equiv 0$ .