2. NOTATION, GENERAL ASSUMPTIONS, AND THE ESTIMATOR

Let {y_k,−∞ < k < ∞} be a GARCH(p,q) process (cf. Bollerslev, 1986) defined by the equations

where ω > 0, α_i ≥ 0, and β_j ≥ 0 are unknown parameters and ε_k, −∞ < k < ∞, are independent and identically distributed random variables. The p ≥ 1 and q ≥ 1 can be any integers. However, by including extra terms with zero coefficients in equation (2.2) we can always achieve the condition min(p,q) ≥ 2, which we shall need in our considerations that follow.

Bougerol and Picard (1992a, 1992b) have found necessary and sufficient conditions for the existence of a unique stationary solution of (2.1) and (2.2) for arbitrary p ≥ 1 and q ≥ 1. Nelson (1991) has solved the problem when p = 1 and q = 1. For the completeness of our presentation, we now state the conditions of Bougerol and Picard (1992a, 1992b). Let

Define the (p + q − 1) × (p + q − 1) matrix A_n, written in block form, by

where I_q−1 and I_p−2 are identity matrices of size q − 1 and p − 2, respectively. The top Lyapunov exponent γ_L associated with the sequence {A_n,−∞ < n < ∞} is

assuming that the expectation

is finite, where the norm

of a matrix M is defined by

with the usual (Euclidean) norm ∥·∥. Bougerol and Picard (1992a, 1992b) show that when

, then equations (2.1) and (2.2) have a unique strictly stationary nonanticipative solution if and only if γ_L < 0. (“Nonanticipative solution” means that {σ_k²,k ≤ i} and {ε_k,k ≥ i} are independent for all −∞ < i < ∞.) Hence, throughout the rest of the present paper we assume γ_L < 0 and work with the aforementioned stationary solution of equations (2.1) and (2.2).

Suppose that the random variables ε_k have a density function (with respect to the Lebesgue measure), which we denote by f. The behavior of f can assist the researcher in finding efficient estimators for the parameters of the GARCH process (cf. Drost and Klaassen, 1997; Berkes and Horváth, 2004). Because f is rarely known in practice, one may need to construct an empirical estimator for the density f and then investigate its statistical properties. Note in this regard that if ε_k were observable, then one could use, for example, the classical kernel-type density estimator

where h = h_n ↓ 0 is the so-called bandwidth and K is the kernel. The function K is usually assumed to be a density function with compact support; we also assume this throughout the present paper.

Because in GARCH models we only observe y₁,…,y_n, the estimator f_n(x) is not practical. To circumvent the problem, we use the residuals (estimated innovations) instead of the unobservable ε₁,…,ε_n. To start with, we use equation (2.1) and write ε_k = y_k /σ_k for every k = 1,…,n. Then we construct estimators

using y₁,…,y_n (we shall discuss this later in this section) and define n new random variables (so-called residuals) by

Finally, we define an estimator

of f by

The main goal of this paper is to present assumptions under which the appropriately normalized L_ν-norm

has a nondegenerate limiting distribution. We note in passing that the results of this paper hold for weighted L_ν-norms also, that is, when dx on the right-hand side of (2.3) is replaced by w(x) dx with any nonnegative bounded function w.

We shall now show that our goal stated in the previous paragraph is actually simpler. Namely, Bickel and Rosenblatt (1973), Csörgő and Horváth (1988), and Beirlant and Mason (1995) have proved that the appropriately centered and scaled νth power of

has asymptotically standard normal distribution. (We refer to these papers for results and assumptions because they are too lengthy to be recorded here.) Hence, the asymptotic behavior of

is established if we show that the quantity

converges to 0 in probability. Interestingly, this result holds when 1 < ν < ∞ but fails when ν = 1. We shall discuss this phenomenon in greater detail in the next section.

We shall now define the aforementioned estimator

of σ_k and in this way complete the definition of

. Let θ = (ω,α₁,…,α_p,β₁,…,β_q) be the vector of all the unknown parameters of the GARCH(p,q) process. We assume that α_i > 0 and β_j > 0. Note that the assumption γ_L < 0 implies β₁ + ··· + β_q < 1. Let ρ, u, and u be such that β₁ + ··· + β_q < ρ < 1 and 0 < u < u ≤ qu < ρ. Define U as the set of the vectors u = (x,s₁,s₂,…,s_p,t₁,t₂,…,t_q) whose all coordinates are not smaller than u, and not larger than u, and such that t₁ + t₂ + ··· + t_q ≤ ρ. We also choose u and u such that θ is in the interior of the set U. (The preceding assumptions are standard in the literature on GARCH models.)

We continue with the well-known (cf., e.g., Berkes, Horváth, and Kokoszka, 2003) equality

where w_k : R^p+q+1 → [0,∞) are (random) functions defined by

The functions c_i(u) : R^p+q+1 → [0,∞) are deterministic and known. For explicit definitions of these functions and their properties we refer to Berkes et al. (2003). In the present paper we use the following property (cf. Berkes et al., 2003, Sect. 3):

where ρ ∈ (0,1) is from the definition of U.

Next, using the observations y₁,…,y_n, we construct an estimator θ_n of θ such that

where ∥·∥ denotes the (usual) Euclidean norm in the space R^1+p+q. We point out that, unless explicitly mentioned otherwise, throughout the current paper we only use property (2.5). Thus, any estimator θ_n satisfying (2.5) could be used. For example, Horváth and Liese (2004) study M-type robust estimators. Drost and Klaassen (1997) and Berkes and Horváth (2004) introduce estimators that are more efficient than the widely used quasi-maximum likelihood estimator.

Replacing θ on the right-hand side of (2.4) by θ_n, we obtain the quantity

The variables

are not of practical use for estimating σ_k because the functions w_k(u) depend on the infinite sequence {y_j : −∞ < j ≤ k − 1}, whereas only y₁,…,y_n are observed and can therefore be used for constructing estimators. To circumvent the problem, we follow Berkes et al. (2003) and introduce truncated versions of the previously defined functions w_k(u) by

The desired estimator

is given by the formula

which completes the rigorous definition of the kernel-type density estimator

. The next section is devoted to asymptotic properties of the estimator

and, in turn, to constructing goodness-of-fit tests.

4. PROOFS

We start with the note that, throughout the proofs, c denotes an unimportant constant whose value may change from line to line. Next we formulate two auxiliary lemmas that we shall constantly use subsequently. When formulating the lemmas, we use the notation

LEMMA 4.1. If E(|ε₀²|^γ) < ∞ for some γ > 0, then E(S₀^κ) < ∞ for any 0 < κ < γ.

Proof. This is Lemma 5.1 in Berkes et al. (2003). █

LEMMA 4.2. If E(|ε₀²|^γ) < ∞ for some γ > 0, then E(S_0,i^κ) < ∞ for any κ > 0 and for both i = 1,2.

Proof. This follows from Lemma 2.3 in Berkes et al. (2003). █

Now we start the proofs of the three theorems. The proofs contain a number of lemmas. The following lemma allows us to reduce our consideration of the norm

to the corresponding properties of

, where

is defined by

with

for all k = 1,…,n.

LEMMA 4.3. If the conditions of Theorem 3.1 are satisfied, then

Proof. Using the mean-value theorem, we have

where ξ_k is a random variable between

. We shall now estimate

. First we write the difference as

Note that

. In the event θ_n ∈ U (and we only need to consider this event because its complement is of negligible probability) we have that c₀(θ_n) ≥ inf_u∈U c₀(u) > 0. Furthermore,

. In view of these notes we have that

does not exceed

. Berkes et al. (2003, Sect. 3) proved that sup_u∈U c_i(u) ≤ cρ^i/q, where ρ ∈ (0,1) is from the definition of U. Hence,

The sum [sum ]_0≤i<∞ ρ^i/qy_−i² is finite almost surely (cf. Berkes et al., 2003). Therefore, from bound (4.3) and equality (4.2) we have that

Using this bound, the Hölder inequality, and also the notation

, we obtain

The integral on the right-hand side of (4.5) cannot be larger than c|h|. This can be checked by first changing the variable of integration x into ξ_k + xh and then using the fact that the first derivative K⁽¹⁾ of K has a bounded support. Therefore,

To complete the proof of statement (4.1), we need to show that the quantity on the right-hand side of (4.6) converges to 0 in probability. Because ν ≥ 1 and nh^3−1/ν → ∞ by assumption, we only need to check that the sum [sum ]_1≤k≤n v_k is of the order

. This follows if the sum [sum ]_1≤k≤n E(v_k) is asymptotically bounded. We estimate the latter sum by

. The moment E(ε_k²) [= E(ε₀²)] is finite. Furthermore, E(S_k) [= E(S₀)] is finite by Lemma 4.1 if E(|ε₀²|^(1+δ)) < ∞ for some δ > 0. The latter moment is finite because we assume E(|ε₀|^{(R−1)max(2,ν)(1+δ)}) < ∞. The sum [sum ]_1≤k≤n ρ^k/q converges because ρ ∈ (0,1). This completes the proof that the right-hand side of (4.6) converges to 0 in probability. The lemma is proved. █

As we already noted, Lemma 4.3 reduces our consideration of the asymptotic behavior of

to the corresponding properties of the norm

. Using the Taylor formula, we have that

where R ≥ 2 is an integer specified in the formulation of Theorem 3.1 and

with ξ_k denoting a random variable between

. It is clear from representation (4.7) that we only need to consider the asymptotics of ∥D_n,r∥_ν and ∥D_n,R*∥_ν. We start with ∥D_n,R*∥_ν because it is the latter term that determines the value of R given in Theorem 3.1.

LEMMA 4.4. If the conditions of Theorem 3.1 are satisfied, then

Proof. Using the Hölder inequality, we have that

The integral ∫_R|K^(R)((x − ξ_k)/h)|^ν dx does not exceed ch because the function K^(R) has a compact support. Consequently, ∥D_n,R*∥_ν does not exceed

. It is natural to expect that the sum

is of the order

. Now taking this statement for granted (cf. (4.9), which follows, for a proof), we obtain that ∥D_n,R*∥_ν ≤ cnh^1/ν. This yields (4.8) provided that nh^{(2R+1−1/ν)/(R−1)} tends to infinity. For the latter statement to hold in view of the assumption nh^A → ∞, we need to have (2R + 1 − 1/ν)/(R − 1) ≤ A or, equivalently, R ≥ (A + 1 − 1/ν)/(A − 2). But this has been assumed. Hence, statement (4.8) holds.

To complete the proof of the lemma, we need to verify that

First we note that

where θ* satisfies ∥θ* − θ∥ ≤ ∥θ_n − θ∥. Hence, we can assume θ* ∈ U without loss of generality because the probability of the opposite event is negligible. It follows from (4.10) that

does not exceed

. Because

does not depend on k and is of the order

, we therefore have (4.9) provided that

. The latter follows if

, which we are now to prove.

The random variable ε_k is independent of S_k and S_k,1. Therefore,

does not exceed the product of E(|ε_k|^R) and E(S_k^R/2S_k,1^R). We estimate the latter expectation using the Hölder inequality. Hence, for any δ₀ > 0, we have that

The right-hand side of (4.11) does not depend on k, and so we only need to check that it is finite. In view of Lemma 4.1, the expectation E(S₀^R(1+δ₀)/2) is finite when E(|ε₀|^R(1+δ)) < ∞ and 0 < δ₀ < δ. The latter expectation is finite because of the assumption E(|ε₀|^{(R−1)max(2,ν)(1+δ)}) < ∞. This also implies that the expectation E(|ε₀|^R) on the right-hand side of (4.11) is finite. To check that the expectation E(S_0,1^R(1+δ)/δ) is finite, we use Lemma 4.2. This finishes the proof that

. Statement (4.9) follows, which also completes the proof of the lemma. █

From Lemma 4.4 we have that

We note that the functions D_n,r(x) are nicer than D_n,R*(x) from a technical point of view. Indeed, the functions D_n,r(x) do not have variables like ξ_k inside them, and therefore they allow us to employ additional technical tools such as martingale theory. Before going into detailed considerations, however, we first simplify statement (4.12) by replacing the functions D_n,r(x) by even simpler functions Δ_n,r(x) defined by

LEMMA 4.5. If the conditions of Theorem 3.1 are satisfied, then for any 1 ≤ r ≤ R − 1 we have that

Proof. Just as in the proof of Lemma 4.4 we showed that ∥D_n,R*∥_ν does not exceed

, we now have that ∥D_n,r − Δ_n,r∥_ν does not exceed

. This time, however, we need a more careful analysis of the difference

than the analysis in the proof of Lemma 4.4. It follows from the Taylor formula that there exists ξ_k satisfying the property ∥ξ_k − θ∥ ≤ ∥θ_n − θ∥ and such that

equals M_k + cn^−1/2R_k with the remainder term

Hence,

does not exceed cn^−1/2(|M_k|^r−1|R_k| + |R_k|^r), which implies the bounds

We note that

, which we can easily verify following the arguments starting with (4.10). Consequently, ∥D_n,r − Δ_n,r∥_ν is of the order

. From this we obtain that (4.13) is a consequence of nh^{2+1/r−1/(rν)} → ∞. Note that, for any r ≥ 1, the quantity 2 + 1/r − 1/(rν) does not exceed 3 − 1/ν. Because nh^3−1/ν → ∞ by assumption, we have nh^{2+1/r−1/(rν)} → ∞. This completes the proof of the lemma. █

From Lemma 4.4 we have the following result:

We note that the first summand (when r = 1) (1/n^3/2h²)Δ_n,1(x) in (4.14) is similar to the function g_n(x), and so we expect it to play a role when ν = 1. Based on this observation, we can now guess how we shall proceed with the following proof. When 1 < ν < ∞, from (4.14) we obtain the statement

We need to show that the right-hand side of (4.15) converges to 0 in probability. When ν = 1, then from (4.14) we have the following statement (note that c_r = 1/r! and thus c₁ = 1):

We need to show that the right-hand side of (4.16) converges to 0 in probability. In summary, we need to show that

for every 1 ≤ r ≤ R − 1 when 1 < ν < ∞, and for every 2 ≤ r ≤ R − 1 when ν = 1.

When proving (4.17), we shall work with the coordinates of the following vectors:

We write

where

Furthermore, denote

We have

It follows from (2.5) that

. Consequently, we obtain from (4.19) that statement (4.17) holds provided that, for any j₁,…,j_r ∈ {1,…,p + q + 1}, we have

Statement (4.20) will turn out to be true for every 1 ≤ r ≤ R − 1 and 1 ≤ ν < ∞ except when both r and ν are equal to 1.

We shall prove (4.20) in two steps. Namely, we shall first prove it (cf. Lemma 4.6, which follows) with g_r,k(x) replaced by E(g_r,k(x)) [= E(g_r,0(x))] and then (cf. Lemma 4.7, which follows) with g_r,k(x) replaced by the centered version

The proof in the first case will be a straightforward mathematical computation, whereas the proof in the second case will be more involved.

LEMMA 4.6. If the conditions of Theorem 3.1 are satisfied, then we have

for every 1 ≤ r ≤ R − 1 and 1 ≤ ν < ∞ except when both r and ν are equal to 1.

Proof. The sum [sum ]_1≤k≤n ν_k(j₁,…,j_r) is of the order

. This fact can be checked by noting that the expected value of |ν_k(j₁,…,j_r)| does not depend on n and is finite provided that the expectations E(|S₀|^r(1+δ₀)/2) and E(|S_0,1|^{r(1+δ₀)/δ₀}) are finite for some δ₀ > 0. We have already proved (cf. the discussion following (4.11)) that the two expectations are finite when r = R. Thus, the expectations are finite for any 1 ≤ r ≤ R − 1.

We shall prove subsequently (cf. (4.23), which follows) that the integral ∫_R|E(g_r,0(x))|^ν dx does not exceed ch^2ν. Hence, statement (4.22) should hold provided that n^r−1h^2r−3+1/ν → ∞. When r = 1, then the latter statement holds indeed because ν > 1 and h → 0. When r ≥ 2, then we write the statement as nh^{(2r−3+1/ν)/(r−1)} → ∞ and note that (2r − 3 + 1/ν)/(r − 1) ≤ 3 − 1/ν for any ν ≥ 1. Because nh^3−1/ν → ∞ by assumption, Lemma 4.6 follows.

To finish the proof of Lemma 4.6, we need to check that

We write the expectation E(g_r,0(x)) as an integral with respect to the density f and then change the variable of integration. This proves that the expectation E(g_r,0(x)) equals h ∫_R(x − zh)^rK^(r)(z) f (x − zh) dz and thus in turn equals h(Γ₁(x) + Γ₂(x)), where

This implies that ∫_R|E(g_r,0(x))|^ν dx does not exceed the sum of ch^ν∫_R|Γ₁(x)|^ν dx and ch^ν∫_R|Γ₂(x)|^ν dx. Hence, we have bound (4.23) provided that, for both i = 1 and i = 2,

To prove (4.24) when i = 1, we first write Γ₁(x) as f (x)∫_R((x − zh)^r − x^r)K^(r)(z) dz. Next, we estimate |(x − zh)^r − x^r| by ch|z|(|x|^r−1 + |z|^r−1). Because K^(r) has a bounded support, |Γ₁(x)| does not exceed chf (x)(1 + |x|^r−1). Consequently,

The integral on the right-hand side of (4.25) is finite by assumption. This concludes the proof of (4.24) when i = 1.

We shall now prove (4.24) when i = 2. We write

Using the Hölder inequality, we have the bounds

The latter bound gives the following one:

Because the integral ∫_R| f⁽¹⁾(x)|^ν(1 + |x|^ν(R−1)) dx is finite and the support of K^(r) is bounded, we obtain from (4.26) that bound (4.24) holds when i = 2. This completes the proof of (4.23) and of the lemma also. █

LEMMA 4.7. If the conditions of Theorem 3.1 are satisfied, then for any values of the indices j₁,…,j_r ∈ {1,…,p + q + 1} we have that

Proof. Let

denote the σ-algebra generated by {ε_k,k ≤ [ell ]}. The stochastic process

is a martingale. Thus, from Theorem 2.11 of Hall and Heyde (1980, p. 23) we have the bound

The right-hand side of the preceding bound equals Ψ₁Ψ₂(x)^ν/2 + cΨ₃(x), where

Therefore, statement (4.27) holds if

To verify (4.28), we need to obtain appropriate bounds for Ψ₁ and for the two integrals with Ψ₂ and Ψ₃. This we do subsequently.

We shall prove that Ψ₁ ≤ cn^ν/2 when 1 ≤ ν < 2. Because ν/2 < 1, in the definition of Ψ₁ we take the expectation E inside {…}^ν/2. This implies that Ψ₁ does not exceed n^p/2{E(ν₀²(j₁,…,j_r))}^ν/2. The expectation E(ν₀²(j₁,…,j_r)) is finite provided that E(S₀^r(1+δ₀)) and E(S_0,1^{2r(1+δ₀)/δ₀}) are finite for some δ₀ > 0. The expectation E(S₀^r(1+δ₀)) is finite because of Lemma 4.1 and the moment assumption E(|ε₀|^{(R−1)2(1+δ)}) < ∞ with δ > δ₀. The expectation E(S_0,1^{2r(1+δ₀)/δ₀}) is finite because of Lemma 4.2. We have therefore proved that Ψ₁ ≤ cn^ν/2 holds when 1 ≤ ν < 2.

We shall now prove that Ψ₁ ≤ cn^ν/2 when ν ≥ 2. Because ν/2 ≥ 1, we need to proceed differently than before. First we note that {[sum ]_1≤k≤n ν_k²(j₁,…,j_r)}^ν/2 does not exceed n^ν/2−1 [sum ]_1≤k≤n|ν_k(j₁,…,j_r)|^ν. Hence, Ψ₁ does not exceed n^ν/2E(|ν₀(j₁,…,j_r)|^ν). This implies that Ψ₁ ≤ cn^ν/2 provided that the expectation E(|ν₀(j₁,…,j_r)|^ν) is finite. The latter follows if the expectations E(S₀^{rν(1+δ₀)/2}) and E(S_0,1^{2rν(1+δ₀)/δ₀}) are finite. The first expectation is finite by Lemma 4.1 due to E(|ε₀|^{(R−1)ν(1+δ)}) < ∞ for some δ > 0. The expectation E(S_0,1^{2rν(1+δ₀)/δ₀}) is finite by Lemma 4.2. This completes the proof of Ψ₁ ≤ cn^ν/2 for any ν ≥ 1.

We shall now estimate the integral ∫_R Ψ₂(x)^ν/2 dx. First we note that Ψ₂(x) does not exceed E(|g_r,0(x)|²). We shall prove in Lemma 4.8 that when α = 2, β = ν/2, and ν ≥ 1, then the integral ∫_R|E(g_r,0²(x))|^ν/2 dx does not exceed ch^ν/2, and so the same upper bound holds for ∫_R Ψ₂(x)^ν/2 dx.

Given the bounds obtained so far, we have that Ψ₁ ∫_R Ψ₂(x)^ν/2 dx does not exceed cn^ν/2h^ν/2. Hence, the quantity in (4.28) with, for a moment, the integral of Ψ₃ deleted converges to 0 provided that nh^2+1/(νr) → ∞. But the latter holds for any r ≥ 1 because nh^2+1/ν → ∞ by assumption. Hence, to finish the proof of statement (4.28), we only need to show that

Using a bound for the expectation of the largest order statistic from Lai and Robbins (1976), we get that, for any d ≥ 1,

(We shall choose d subsequently.) Next, we express the expectation on the right-hand side of (4.30) as follows:

where the σ-algebra

is the same as before. We shall now discuss the expectations on the right-hand side of (4.31) in detail.

The first expectation on the right-hand side of (4.31) is finite provided that the two expectations E(S₀^{rνd(1+δ₀)/2}) and E(S_0,1^{rνd(1+δ₀)/δ₀}) are finite for some δ₀ > 0. The finiteness of E(S_0,1^{rνd(1+δ₀)/δ₀}) follows from Lemma 4.2 independently of the value of d ≥ 1. The finiteness of E(S₀^{rνd(1+δ₀)/2}) depends on d. Therefore, we now state the choice of d that will be suggested by our calculations that follow. Namely, we have

If d = 1, then the expectation E(S₀^{rνd(1+δ₀)/2}) equals E(S₀^{rν(1+δ₀)/2}). We have already verified that the latter expectation is finite because E(|ε₀|^{(R−1)ν(1+δ)}) < ∞ for some δ > 0, when ν ≥ 2. If d = 2/ν, then E(S₀^{rνd(1+δ₀)/2}) equals E(S₀^r(1+δ₀)). We already proved that the latter expectation is finite because E(|ε₀|^{(R−1)2(1+δ)}) < ∞ for some δ > 0, when 1 ≤ ν < 2.

In view of the notes in the previous paragraph and the fact that the expectation E(|G_r,0(x)|^νd) does not exceed cE(|g₀(x)|^νd), we have from (4.30) and (4.31) that the function Ψ₃(x) does not exceed cn^1/d(E(|g_r,0(x)|^νd))^1/d. This in turn implies that ∫_R Ψ₃(x) dx does not exceed cn^1/d∫_R|E(|g_r,0(x)|^νd)|^1/d dx. With the notations α = νd and β = 1/d, the latter integral equals ∫_R|E(|g_r,0(x)|^α)|^β dx, which we denote by Ξ for brevity.

When 1 ≤ ν < 2, then d = ν/2 and thus α = 2 and β = ν/2. Lemma 4.8, which follows, implies that Ξ ≤ ch^1/d. When ν ≥ 2, then d = 1 and thus α = ν and β = 1. Lemma 4.8 again implies the bound Ξ ≤ ch^1/d. Hence, the integral ∫_R Ψ₃(x) dx does not exceed cn^1/dh^1/d. This implies (4.29) provided that nh^{1+(r+1/ν)/(r+1−2/(νd))} → ∞. The latter follows if the exponent of h does not exceed A. This is equivalent to

Recall that d has to be at least 1. Because r ≥ 1 and A ≥ 2 + 1/ν by assumption, the ratio in the parentheses on the right-hand side of (4.33) does not exceed 1. Thus, the choice of d given in (4.32) is indeed natural and satisfies condition (4.33). This completes the proof of (4.29) and of the lemma also. █

In the proof of Lemma 4.7 we referred several times to Lemma 4.8. Here now is its formulation, followed by the proof.

LEMMA 4.8. If the conditions of Theorem 3.1 are satisfied, then we have that

for the following sets of parameters:

Proof. We start with the bound

where C > 0 is a finite constant such that that K^(r) is 0 outside the interval [−C,C]. Hence, ∫_R|E(|g_r,0(x)|^α)|^β dx does not exceed ch^βI(α,β), where

The lemma is proved if we show that the integral I(α,β) is finite.

When β ≥ 1, then the integral does not exceed c ∫_R|x|^rαβf (x)^β dx. Note that when β ≥ 1, then we have two cases: first, α = 2, β = ν/2, and ν ≥ 2; second, α = ν, β = 1, and ν ≥ 2. In the first case the integral ∫_R|x|^rαβf (x)^β dx is equal to ∫_R|x|^νrf (x)^ν/2 dx, and in the second case it is equal to ∫_R|x|^νrf (x) dx. The integral ∫_R|x|^νrf (x)^ν/2 dx does not exceed the sum of the integrals ∫_R|x|^νrf (x) dx and ∫_R|x|^νrf (x)^ν dx. The first integral equals E(|ε₀|^νr) < ∞, which is finite as we have already noted earlier. The integral ∫_R|x|^νrf (x)^ν dx is finite by assumption.

When 0 < β < 1, we choose any γ > 1 − β and using the Hölder inequality obtain that

where u = 1/(1 − β), v = 1/β, and

The first integral on the right-hand side of (4.36) is finite. To prove that the integral I_γ(α,β) is also finite, we obtain the bound

The integral on the right-hand side of (4.37) is finite provided that E(|ε₀|^rα+γ/β) < ∞. Because 0 < β < 1, we only need to consider the case when α = 2, β = ν/2, and 1 ≤ ν < 2. Hence, the integral on the right-hand side of (4.37) is finite provided that E(|ε₀|^{(2r−1+2/ν)(1+δ)}) < ∞ for some δ > 0. Note that 2r − 1 + 2/ν does not exceed 2R − 1. Hence, the expectation E(|ε₀|^{(2r−1+2/ν)(1+δ)}) is finite because E(|ε₀|^{(2R−1)(1+δ)}) < ∞ when 1 ≤ ν < 2 by assumption. █

With the preceding lemma we have completed the verification of statement (4.17). In view of (4.15), we therefore have that

for every 1 < ν < ∞. In view of (4.16) we have that, when ν = 1,

We already noted that (1/n^3/2h²)Δ_n,1(x) is very similar to the function g_n(x). We now need to establish a more precise relationship between the two random functions. We write the equalities

From (4.40) and (4.41) we have that

where G_1,k(x) is defined in (4.21). Lemma 4.7 implies that, for every ν ≥ 1,

Statements (4.38), (4.39), and (4.42) complete the proof of Theorem 3.1. Statements (4.38) and (4.42) complete the proof of Theorem 3.2. To complete the proof of Theorem 3.3, we need an additional argument.

We write the equality

Because the random variables w_k⁽¹⁾(θ)/w_k(θ), k ≥ 1, make a stationary and ergodic sequence with finite means (cf. Berkes et al., 2003), by the ergodic theorem we have that n⁻¹ [sum ]_1≤k≤n w_k⁽¹⁾(θ)/w_k(θ) converges to E(w₀⁽¹⁾(θ)/w₀(θ)) almost surely. Next we show that

Before starting the proof of (4.44) we note that the preceding results imply Theorem 3.3 because of the convergence in distribution of

to ξ.

To prove (4.44), we use the arguments following (4.23) with ν = 1 and r = 1, and we express the expectation E(g_1,0(x)) as the sum of hΓ₁(x) and hΓ₂(x), where Γ₁(x) = hf (x) and

Hence,

where

From (4.45) we have (4.44) provided that the integrals ∫_R|Γ₂*(x)|dx and ∫_R|Γ₂**(x)|dx converge to 0 when h → 0. Using same arguments as those in (4.26), we have that ∫_R|Γ₂**(x)|dx converges to 0 provided that | f⁽¹⁾(x)| is integrable, which follows from the assumptions of Theorem 3.1. Using similar arguments and also the Lebesgue dominated convergence theorem, we obtain that ∫_R|Γ₂*(x)|dx converges to 0 provided that the function (1 + |x|)| f⁽¹⁾(x)| is integrable, which follows from the assumptions of Theorem 3.1. This finishes the proof of Theorem 3.3.