Almost sure limit theorems for the maximum of stationary Gaussian sequences

Endre Csáki^{a,}^{1}, Khurelbaatar Gonchigdanzan^{b,}^{2}

aA. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O.Box 127, H-1364, Budapest, Hungary bDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA

Abstract. We prove an almost sure limit theorem for the maxima of stationary Gaussian sequences with covariance r_{n} under the condition r_{n} log n (log log n)^{1+ε} = O(1). Key words: almost sure central limit theorem, logarithmic average, stationary Gaussian sequence.

Introduction. The early results on the almost sure central limit theorem (ASCLT) dealt mostly with partial sums of random variables. A general pattern of these investigations is that if X_{1}, X_{2}, … is a sequence of random variables with partial sums S_{n} = ∑_{k=1}^{n} X_{k} satisfying a_{n}(S_{n} − b_{n}) ⟶^{D} G for some numerical sequences (a_{n}), (b_{n}) and distribution function G, then under some additional mild conditions we have

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} I (a_{k}(S_{k} − b_{k}) < x) = G(x) a.s.

for any continuity point x of G, where I is indicator function.

For more discussions about ASCLT we refer to the survey papers by Berkes (1998), and Atlagh and Weber (2000). Recently Fahrner and Stadtmüller (1998) and Cheng et al. (1998) have extended this principle by proving ASCLT for the maxima of independent random variables.

Typeset by AMS-TEX

1Supported by the Hungarian National Foundation for Scientific Research Grant No. T 029621

2Supported by a TAFT Fellowship at the University of Cincinnati

THEOREM A. Let X_{1}, X_{2}, … be i.i.d. random variables and M_{k} = max_{i≤k} X_{i}. If a_{k}(M_{k} − b_{k}) ⟶^{D} G for a nondegenerate distribution G and some numerical sequences (a_{k}) and (b_{k}) , then we have

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} I (a_{k}(M_{k} − b_{k}) < x) = G(x) a.s.

for any continuity point x of G.

Berkes and Csáki (2001) extended the ASCLT for general nonlinear functionals of independent random variables. For strong invariance principles improving Theorem A see Berkes and Horváth (2001) and Fahrner (2001).

Throughout this paper Z_{1}, Z_{2}, … is a stationary Gaussian sequence and we denote its covariance function by r_{n} = Cov(Z_{1}, Z_{n+1}), and M_{n} = max_{1≤i≤n} Z_{i} and M_{k,n} = max_{k+1≤i≤n} Z_{i}. Here a ≪ b and a ∼ b stand for a = O(b) and a/b → 1 respectively. Φ(x) is the standard normal distribution function and ϕ(x) is its density function.

For notational convenience let R(n) = r_{n} log n (log log n)^{1+ε}.

1. Main Result. The main result is an almost sure central limit theorem for the maximum of stationary Gaussian sequences. THEOREM 1.1. Let Z_{1}, Z_{2}, … be a standardized stationary Gaussian sequence with R(n) = O(1) as n → ∞. Then

(i) If n(1 − Φ(u_{n})) → τ for 0 ≤ τ < ∞, then

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} I(M_{k} ≤ u_{k}) = e^{−τ} a.s.,

(ii) If a_{n} = (2 log n)^{1/2} and b_{n} = (2 log n)^{1/2} − \frac{1}{2}(2 log n)^{−1/2}(log log n + log 4π), then

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} I(a_{k}(M_{k} − b_{k}) ≤ x) = exp(−e^{−x}) a.s..

2. Auxiliary Results. The main weak convergence result for the maximum of stationary Gaussian sequence is summarized in the following theorem.

THEOREM 2.1. (Theorem 4.3.3 in Leadbetter et al. (1983)). Let Z_{1}, Z_{2}, … be a standardized stationary Gaussian sequence with r_{n} log n → 0. Then

(i) For 0 ≤ τ < ∞, P(M_{n} ≤ u_{n}) → e^{−τ} if and only if n(1 − Φ(u_{n})) → τ (ii) P(a_{n}(M_{n} − b_{n}) ≤ x) → exp(−e^{−x}), where a_{n} = (2 log n)^{1/2} and b_{n} = (2 log n)^{1/2} − \frac{1}{2}(2 log n)^{−1/2}(log log n + log 4π).

We need the following lemmas for the proof of our main result. LEMMA 2.1. Let Z_{1}, Z_{2}, … be a standardized stationary Gaussian sequence. Assume that R(n) = O(1) and n(1 − Φ(u_{n})) is bounded. Then

sup_{1≤k≤n} k ∑_{j=1}^{n} |r_{j}| exp (−\frac{u_{k}^{2} + u_{n}^{2}}{2(1 + |r_{j}|)}) ≪ (log log n)^{−(1+ε)}.

PROOF OF LEMMA 2.1: Under the condition r_{n} → 0 we have sup_{n≥1} |r_{n}| = σ < 1 (cf., Leadbetter et al., 1983). By assumption, n(1 − Φ(u_{n})) ≤ K. Let the sequence (v_{n}) be defined by v_{n} = u_{n} if n ≤ K and n(1 − Φ(v_{n})) = K, if n > K. Then clearly u_{n} ≥ v_{n} and hence

k ∑_{j=1}^{n} |r_{j}| exp (−\frac{u_{k}^{2} + u_{n}^{2}}{2(1 + |r_{j}|)}) ≤ k ∑_{j=1}^{n} |r_{j}| exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2(1 + |r_{j}|)}) .

Thus it would be enough to prove the lemma for the sequence (v_{n}). By the well known fact

1 − Φ(x) ∼ \frac{ϕ(x)}{x}, x → ∞

we can see that

(2.1)  exp (−\frac{v_{n}^{2}}{2} ) ∼ \frac{K\sqrt{2π}v_{n}}{n} ,  v_{n} ∼ (2 log n)^{1/2}.

Define α to be 0 < α < (1 − σ)/(1 + σ). Note that

k ∑_{j=1}^{n} |r_{j}| exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2(1 + |r_{j}|)}) =

= k ∑_{1≤j≤n^{α}} |r_{j}| exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2(1 + |r_{j}|)}) + k ∑_{n^{α}<j≤n} |r_{j}| exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2(1 + |r_{j}|)}) =

=: T_{1} + T_{2}.

Using (2.1)

T_{1} ≤ kn^{α} exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2(1 + σ)}) = kn^{α} (exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2} ))^{1/(1+σ)} ≪

≪ kn^{α} (\frac{v_{k}v_{n}}{kn} )^{1/(1+σ)} ≪ k^{1−1/(1+σ)}n^{α−1/(1+σ)}(log k log n)^{1/2(1+σ)} ≤

≤ n^{1+α−2/(1+σ)}(log n)^{1/(1+σ)}.

Since 1 + α − 2/(1 + σ) < 0, we get T_{1} ≤ n^{−δ} for some δ > 0, uniformly for 1 ≤ k ≤ n. Now we estimate the second term T_{2}. Setting σ_{n} = sup_{j≥n} |r_{j}| and counting on R(n) = O(1) as n → ∞

(2.2)  σ_{n} log n(log log n)^{1+ε} ≤ sup_{j≥n} |r_{j}| log j(log log j)^{1+ε} = O(1), n → ∞.

Set p = [n^{α}]. By (2.1) and (2.2) we have

σ_{p}v_{k}v_{n} ≪ σ_{[n^{α}]}(log k log n)^{1/2} ≪ σ_{[n^{α}]} log n^{α} ≪

(2.3) ≪ (log log n^{α})^{−(1+ε)} ∼ (log log n)^{−(1+ε)}

and similarly, for 1 ≤ k ≤ n

(2.4)  σ_{p}v_{k}^{2} ≪ (log log n)^{−(1+ε)}.

Hence using (2.1), (2.3) and (2.4)

T_{2} ≤ kσ_{p} exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2} ) ∑_{p≤j≤n} exp (\frac{(v_{k}^{2} + v_{n}^{2})|r_{j}|}{2(1 + |r_{j}|)}) ≤

≤ knσ_{p} exp (−\frac{v_{k}^{2} + v_{n}^{2}}{2} ) exp (\frac{(v_{k}^{2} + v_{n}^{2})σ_{p}}{2} ) ≪ (log log n)^{−(1+ε)}.

The proof is completed.

LEMMA 2.2. Let Z_{1}, Z_{2}, … be a standard stationary Gaussian sequence. Suppose that sup_{n≥1} |r_{n}| < 1. Then for k < n

|P (M_{k} ≤ u_{k}, M_{k,n} ≤ u_{n}) − P (M_{k} ≤ u_{k})P (M_{k,n} ≤ u_{n})| ≪

≪ k ∑_{j=1}^{n} |r_{j}| exp (−\frac{u_{k}^{2} + u_{n}^{2}}{2(1 + |r_{j}|)}) .

PROOF OF LEMMA 2.2. We use the following THEOREM 2.2. (Theorem 4.2.1, Normal Comparison Lemma in Leadbetter et al. (1983)). Suppose ξ_{1}, …, ξ_{n} are standard normal variables with covariance matrix Λ^{1} = (Λ_{ij}^{1}), and η_{1}, …, η_{n} similarly with covariance matrix Λ^{0} = (Λ_{ij}^{0}), and let ρ_{ij} = max(|Λ_{ij}^{1}|, |Λ_{ij}^{0}|). Further, let u_{1}, …, u_{n} be real numbers. Then

|P (ξ_{j} ≤ u_{j}, j = 1, …, n) − P (η_{j} ≤ u_{j}, j = 1, …, n)| ≤

≤ K ∑_{1≤i<j≤n} |Λ_{ij}^{1} − Λ_{ij}^{0}| exp (−\frac{u_{i}^{2} + u_{j}^{2}}{2(1 + ρ_{ij})}) .

Apply this Theorem with (ξ_{i} = Z_{i}, i = 1, …, n), (η_{j} = Z_{j}, j = 1, …, k; η_{j} = Z̃_{j}, j = k + 1, …, n), where (Z̃_{k+1}, …, Z̃_{n}) has the same distribution as (Z_{k+1}, …, Z_{n}), but it is independent of (Z_{1}, …, Z_{k}). Further, u_{i} = u_{k}, i = 1, …, k and u_{i} = u_{n}, i = k + 1, …, n. Then Λ_{ij}^{1} = Λ_{ij}^{0} = r_{j−i} if either 1 ≤ i < j ≤ k, or k + 1 ≤ i < j ≤ n. Otherwise Λ_{ij}^{1} = r_{j−i}, Λ_{ij}^{0} = 0. Hence we have

|P (M_{k} ≤ u_{k}, M_{k,n} ≤ u_{n}) − P (M_{k} ≤ u_{k})P (M_{k,n} ≤ u_{n})| ≪

≪ ∑_{i=1}^{k} ∑_{j=k+1}^{n} |r_{j−i}| exp (−\frac{u_{k}^{2} + u_{n}^{2}}{2(1 + |r_{j−i}|)}) ≤ k ∑_{m=1}^{n} |r_{m}| exp (−\frac{u_{k}^{2} + u_{n}^{2}}{2(1 + |r_{m}|)}) .

This completes the proof of LEMMA 2.2.

LEMMA 2.3. Let Z_{1}, Z_{2}, … be a standardized stationary Gaussian sequence. Assume that R(n) = O(1) and n(1 − Φ(u_{n})) is bounded. Then for 1 ≤ k < n

Cov (I(M_{k} ≤ u_{k}), I(M_{k,n} ≤ u_{n})) ≪ (log log n)^{−(1+ε)}.

PROOF OF LEMMA 2.3: It follows simply from LEMMA 2.1 and LEMMA 2.2.

LEMMA 2.4. Let Z_{1}, Z_{2}, … be a standardized stationary Gaussian sequence. Assume that R(n) = O(1) and n(1 − Φ(u_{n})) is bounded, then

E|I(M_{n} ≤ u_{n}) − I(M_{k,n} ≤ u_{n})| ≪ \frac{k}{n} + (log log n)^{−(1+ε)}.

PROOF OF LEMMA 2.4: Note that

E|I(M_{n} ≤ u_{n}) − I(M_{k,n} ≤ u_{n})| = P(M_{k,n} ≤ u_{n}) − P(M_{n} ≤ u_{n}) ≤

≤ |P(M_{k,n} ≤ u_{n}) − Φ^{n−k}(u_{n})| + |P(M_{n} ≤ u_{n}) − Φ^{n}(u_{n})|+

+ |Φ^{n−k}(u_{n}) − Φ^{n}(u_{n})| =: D_{1} + D_{2} + D_{3}.

From the elementary fact that

x^{n−k} − x^{n} ≤ \frac{k}{n}, 0 ≤ x ≤ 1

we have D_{3} ≤ (k/n). By Corollary 4.2.4 in Leadbetter et al. (1983), p. 84

D_{i} ≪ n ∑_{j=1}^{n} |r_{j}| exp (−\frac{u_{n}^{2}}{1 + |r_{j}|}) i = 1, 2.

Thus by LEMMA 2.1 we have D_{i} ≪ (log log n)^{−(1+ε)}, i = 1, 2.

3. Proof of Main Result. We now give the proof of THEOREM 1.1. We need the following lemma for the proof. LEMMA 3.1. Let η_{1}, η_{2}, … be a sequence of bounded random variables. If

Var (∑_{k=1}^{n} \frac{1}{k} η_{k}) ≪ log^{2}n (log log n)^{−(1+ε)} for some ε > 0,

then

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} (η_{k} − Eη_{k}) = 0 a.s..

PROOF OF LEMMA 3.1: Setting

μ_{n} = \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} (η_{k} − Eη_{k})

and n_{k} = exp(exp(k^{ν})) for some \frac{1}{1+ϵ} < ν < 1, we have

∑_{k=3}^{∞} Eμ_{n_{k}}^{2} ≪ ∑_{k=3}^{∞}(log log n_{k})^{−(1+ϵ)} ≪ ∑_{k=3}^{∞} k^{−ν(1+ϵ)} < ∞

implying ∑_{k=3}^{∞} μ_{n_{k}}^{2} < ∞ a.s. Thus

μ_{n_{k}} → 0 a.s..

Since

(k + 1)^{ν} − k^{ν} → 0 as k → ∞ if ν < 1,

we have

\frac{log n_{k+1}}{log n_{k}} = e^{(k+1)^{ν}−k^{ν}} → 1 as k → ∞.

Obviously for any given n there is an integer k such that n_{k} < n ≤ n_{k+1}. Therefore

|μ_{n}| ≤ \frac{1}{log n} |∑_{j=1}^{n} \frac{1}{j} (η_{j} − Eη_{j})| ≤

≤ \frac{1}{log n_{k}} |∑_{j=1}^{n_{k}} \frac{1}{j} (η_{j} − Eη_{j})| + \frac{1}{log n_{k}} ∑_{j=n_{k}+1}^{n_{k+1}} \frac{1}{j} |η_{j} − Eη_{j}| ≪

≪ |μ_{n_{k}}| + \frac{1}{log n_{k}} (log n_{k+1} − log n_{k}) ≪ |μ_{n_{k}}| + (\begin{matrix}log & n_{k+1} \\ log & n_{k} \end{matrix} − 1)

and thus

lim_{n→∞} μ_{n} = 0 a.s..

PROOF OF THEOREM 1.1: First, we claim that under the assumptions that R(n) = O(1) and n(1 − Φ(u_{n})) is bounded, we have

(3.1)  lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} (I(M_{k} ≤ u_{k}) − P(M_{k} ≤ u_{k})) = 0 a.s..

In order to show this, by LEMMA 3.1 it is sufficient to show

(3.2) Var (∑_{k=1}^{n} \frac{1}{k} I(M_{k} ≤ u_{k})) ≪ (log log n)^{−(1+ε)} log^{2} n for some ε > 0.

Let η_{k} = I(M_{k} ≤ u_{k}) − P(M_{k} ≤ u_{k}). Then

Var(∑_{k=1}^{n} \frac{1}{k} I(M_{k} ≤ u_{k})) = E(∑_{k=1}^{n} \frac{1}{k} η_{k})^{2} =

= ∑_{k=1}^{n} \frac{1}{k^{2}} E|η_{k}|^{2} + 2 ∑_{1≤k<l≤n} \frac{|E(η_{k}η_{l})|}{kl} =: L_{1} + L_{2}. (3.3)

Since |η_{k}| ≤ 2, it follows that

(3.4)  L_{1} ≪ ∑_{k=1}^{∞} \frac{1}{k^{2}} < ∞.

To estimate L_{2}, note that for l > k

|E(η_{k}η_{l})| = |Cov(I(M_{k} ≤ u_{k}), I(M_{l} ≤ u_{l}))| ≤ |Cov(I(M_{k} ≤ u_{k}), I(M_{l} ≤ u_{l})−
− I(M_{k,l} ≤ u_{l}))| + |Cov(I(M_{k} ≤ u_{k}), I(M_{k,l} ≤ u_{l}))| ≪

(3.5) ≪ E|I(M_{l} ≤ u_{l}) − I(M_{k,l} ≤ u_{l})| + |Cov(I(M_{k} ≤ u_{k}), I(M_{k,l} ≤ u_{l}))|.

By LEMMA 2.3 and LEMMA 2.4 we get

|Cov(I(M_{k} ≤ u_{k}), I(M_{k,l} ≤ u_{l}))| ≪ (log log l)^{−(1+ε)}

and

E|I(M_{l} ≤ u_{l}) − I(M_{k,l} ≤ u_{l})| ≪ \frac{k}{l} + (log log l)^{−(1+ε)}.

Hence for l > k

(3.6)  |E(η_{k}η_{l})| ≪ \frac{k}{l} + (log log l)^{−(1+ε)}

and consequently

L_{2} ≪ ∑_{1≤k<l≤n} \frac{1}{kl} (\frac{k}{l} ) + ∑_{1≤k<l≤n} \frac{1}{kl(log log l)^{1+ε}} =

=: L_{21} + L_{22}. (3.7)

For L_{21} and L_{22} we have the following estimates:

L_{22} ≪ ∑_{l=3}^{n} \frac{1}{l(log log l)^{1+ε}} ∑_{k=1}^{l−1} \frac{1}{k} ≪ ∑_{l=3}^{n} \frac{log l}{l(log log l)^{1+ε}} ≪

(3.8) ≪ log n ∑_{l=3}^{n} \frac{1}{l(log log l)^{1+ε}} ≪ log^{2} n (log log n)^{−(1+ε)}

and

(3.9)  L_{21} ≤ ∑_{1≤k<l≤n} \frac{1}{kl} (\frac{k}{l} ) ≪ log n.

Thus (3.3)–(3.9) together establish (3.1). PROOF OF (i): Note that R(n) = O(1) implies r_{n} log n → 0. By THEOREM 4.3.3(i) in Leadbetter et al. (1983), we have P (M_{n} ≤ u_{n}) → e^{−τ}. Clearly this implies

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} P(M_{k} ≤ u_{k}) = e^{−τ}

which is, by (3.1), equivalent to

lim_{n→∞} \frac{1}{log n} ∑_{k=1}^{n} \frac{1}{k} I(M_{k} ≤ u_{k}) = e^{−τ} a.s..

PROOF OF (ii): By THEOREM 2.1 we have n(1 − Φ(u_{n})) → e^{−x} for u_{n} = x/a_{n} + b_{n}. Thus the statement of (ii) is a special case of (i).

Acknowledgement

We thank a referee for some useful comments.

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