On the almost sure convergence of randomly indexed maximum of random variables

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXXIII, NO. 2, 2019 SECTIO A 91–104

ANDRZEJ KRAJKA, ZDZISŁAW RYCHLIK and JOANNA WASIURA-MAŚLANY

On the almost sure convergence of randomly indexed maximum

of random variables

Dedicated to Professor Yuri Kozitsky on the occasion of his 70th birthday

Abstract. We prove an almost sure random version of a maximum limit theorem, using logarithmic means for max1≤i≤NnXi, where {Xn, n ≥ 1} is a sequence of identically distributed random variables and {Nn, n ≥ 1} is a sequence of positive integer random variables independent of {Xn, n ≥ 1}.

Furthermore, we consider the almost sure random version of a limit theorem for kth order statistics.

1. Introduction

Let {X_n, n ≥ 1} be a sequence of independent and identically distributed random variables with EX₁ = 0, EX₁² = 1, and let S_n= X₁+X₂+· · ·+X_n. The almost sure central limit theorem (ASCLT) says that for any fixed x ∈ <

we have

(1) lim

n→∞

1 log n

n

X

j=1

1 jI S_j

√j ≤ x



= Φ(x), a.s.,

where Φ(x) denotes the standard normal distribution function. This result is a generalization of the arcsin law of Andersen and was firstly obtained by

2010 Mathematics Subject Classification. Primary: 60F05; Secondary: 60F15, 60G70.

Key words and phrases. Almost sure central limit theorem, randomly indexed sums.

Brosamler [4] and Schatte [19] under additional moment conditions on X₁ and by Lacey and Philipp [13] under assuming only finite variance. This result is probably the most intensively investigated in the last decade. For the different generalizations of (1) cf. [3].

Let us consider the following three sets of distribution functions:

Case (i) D₁ = n

F ∈ L : there exists the positive function g such that

1−F (t+xg(t))

1−F (t) −→ e^−x, as t → x_F, for all x ∈ <o , Case (ii) D_2,α = n

F ∈ L : x_F = ∞ and ^{1−F (tx))}_{1−F (t)} −→ x^−α, as t → ∞, for all x > 0o

, for some α > 0, Case (iii) D_3,α = n

F ∈ L : x_F < ∞ and ^{1−F (x}_{1−F (x}^F−xh)

F−h) −→ x^α, as h → 0₊, for all x > 0o

, for some α > 0,

where F (.) denotes the distribution function of X₁, x_F = inf{x : F (x) = 1}, and L denotes the set of distribution functions on <. It is known (cf. [14, 17]) that if F belongs to D₁, D_2,αor D_3,αwith some α > 0, then there exist con- stants {a_n, bn, n ≥ 1} such that

(2) an



1≤j≤nmax Xj− b_n

 D

−→ G, as n → ∞, where G is equal to

G1(x) = e^−e−x, G2,α(x) =

(0, x ≤ 0, e^−x−α, x > 0, or

G_3,α(x) =

(e^−(−x)α, x ≤ 0, 1, $x > 0,

respectively. Conversely, if (2) holds for some sequence of independent and identically distributed random variables {X_n, n ≥ 1}, then the possible nondegenerate limits G are G₁, G_2,α, or G_3,α only. Furthermore, under assumption (2) we have

(3) a_n(X_n−k:n− b_n)−→ G(x)^D

k

X

t=0

(− log G(x))^t

t! , as n → ∞,

where by X_1:n≤ X_2:n≤ X_3:n≤ · · · ≤ X_n:n we denote the order statistics of {X₁, X₂, . . . , X_n}. These results are called the max limit theorems. In 1998, Fahrner I. and Stadtm¨uller V. [8], and independently Cheng S., Peng L. and Qi Y. [6] proved the max limit schema version of ASCLT with k = 0 (cf. [7],

too). They proved that if (2) holds, then

(4) lim

n→∞

1 log n

n

X

j=1

1 jI



aj( max

1≤i≤jXi− b_j) ≤ x



= G(x), a.s.

for any x ∈ <. This result was generalized on case kth order statististic by Stadtm¨uller [18]. Assuming (2) and that {X_n, n ≥ 1} is an independent and identically distributed sequence with continuous distribution function of X₁, he proved that

(5) lim

n→∞

1 log n

n

X

j=1

1

jI[a_j(X_j−k:j − b_j) ≤ x] = G(x)

k

X

t=0

(− log G(x))^t t! , a.s.

(In order to get simpler formulas here and in what follows, we put P [X_j:k ≤ x] := 1 for j ≤ 0, k ≥ 0 or k > j.) However, among the different generaliza- tions of ASCLT there is no version of ASCLT with random indices, although the first central limit theorem results and max limit theorem results almost at once obtained such generalization (cf., for e.g., [16], in CLT case and [1]

in max limit theorem case). The main reason is that the random indexing introduces the big level of complications and numerical difficulties. In this paper we generalize the results of [6, 8] and [18] in the following directions:

(i) We drop the assumption of interindependency of {X_n, n ≥ 1} con- sidering the stationary sequences.

(ii) We consider the randomly indexed version of (5). Assuming inde- pendence between the sequence {X_n, n ≥ 1} and the sequence of random indices {N_n, n ≥ 1}, we give the conditions under which

(6)

n→∞lim 1 log n

n

X

j=1

1 j



I[a_j(X_Nj−k:N_j− b_j) ≤ x]

− G^Njj (x)

k

X

t=0

1 t!

h

−Nj

j log G(x) it

= 0, a.s.

(iii) In comparison with the result of [18], we omit the assumption on continuity of the distribution function of X₁.

In the whole paper we will use the notations: x ∨ y = max{x, y} and x ∧ y = min{x, y}.

2. Main results

Let {X_n, n ≥ 1} be a sequence of identically distributed random variables with the common distribution function F such that for some constants {a_n, b_n, n ≥ 1} we have

(7) a_n



1≤j≤nmax X_j− b_n

 D

−→ G, as n → ∞,

with G equal to G₁, G_2,α or G_3,α. Let { ˜X_n, n ≥ 1} be a sequence of inde- pendent and identically distributed random variables with the distribution function F . For x, y ∈ < we will put

v_j = x/a_j+ b_j, j ≥ 1,

and for some positive constants θ and positive integer k:

α_j,h(x, y) = |P [X_j−k:j ≤ x, X_h−k:h≤ y] − P [ ˜X_j−k:j ≤ x, ˜X_h−k:h≤ y]^θ|, α_j(x) = |P [X_j−k:j ≤ x] − P [ ˜X_j−k:j ≤ x]^θ|.

(8)

The coefficients α_j(x) defined in (8) are called the extremal index of sta- tionary sequence {X_n, n ≥ 1} and were introduced in [15] and studied in- tensively in [11]. These coefficients stand the analogue of mixing condition in max-limit theory.

Theorem 1. Let {X_n, n ≥ 1} be a sequence of identically distributed ran- dom variables with common distribution function F satisfying condition (7) for some numbers {a_n, b_n, n ≥ 1}. Let {N_n, n ≥ 1} be a sequence of pairwise independent random indexes independent of {X_n, n ≥ 1}. Let us assume that for some fixed µ ∈ (0, 1),

n

X

h=1 h−1

X

j=1

1

jhE N_j ∧ N_h

h ∧ 1



= O (log n)^2−µ
. (9)

Furthermore, let us assume that

n

X

h=1 h−1

X

j=1

1

jhEα_Nj,Nh(vj, v_h) = O (log n)^2−µ
, (10)

and

n

X

j=1

1

jEαNj(vj) = O (log n)^2−µ
. (11)

In the case when G = G_2,α with some α > 0, we assume additionally that for some δ₀ > 0,

P N_j j < δ0



= O (log j)^−µ
, as j → ∞.

(12) Then

(13) 1

log n

n

X

j=1

1 j



Ia_j(X_Nj−k:N_j− b_j) ≤ x − H_G,k,N^θ

j/j(x) _a.s.

−→ 0, as n → ∞, where

HG,k,β(x) = (

G^β(x)Pk t=0 1

t![−β log G(x)]^t, if G(x) > 0,

0, if G(x) = 0.

Let f (.) be a a.e. continuous, bounded real function, such that f (−∞) = 0, f (+∞) = 0. If lim inf ^N_j^j > C > 0, then

(14) 1 log n

n

X

j=1

1 j



f aj(X_Nj−k:Nj− b_j)
− Z ∞

−∞

f (x)H_G,k,N^θ

j/j(dx)



−→ 0,a.s.

as n → ∞.

Additionally, if there exists a positive bounded from 0 random variable λ such that d

_N

j

j , λ



= O((log j)^−µ), where d(X, Y ) is the L´evy–Prokhorov’s distance between random variables X and Y (i.e. d(X, Y ) = inf{ > 0 : P [|X − Y | > ] < }), then

(15) 1 log n

n

X

j=1

1 j



f aj(X_Nj−k:Nj− b_j)
− Z ∞

−∞

f (x)H_G,k,λ^θ (dx)



−→ 0,a.s.

as n → ∞.

Corollary 1. Let {X_n, n ≥ 1} be a sequence of independent and identically distributed random variables with common distribution function F, and let {N_n, n ≥ 1} be a sequence of pairwise independent random indexes indepen- dent of {X_n, n ≥ 1}. Let us assume (7), (9), and in case when G = G2,α

with some α > 0, (12) hold. Then

1 log n

n

X

j=1

1 j



Ia_j(XNj−k:Nj− b_j) ≤ x − H_G,k,Nj/j(x)

 a.s.

−→ 0, as n → ∞.

Let f (.) be a a.e. continuous, bounded real function, such that f (−∞) = 0, f (+∞) = 0. If lim inf ^N_j^j > C > 0, then

1 log n

n

X

j=1

1 j



f aj(XNj−k:Nj− b_j)
− Z ∞

−∞

f (x)H_G,k,Nj/j(dx)

 a.s.

−→ 0, as n → ∞.

Additionally, if there exists a positive bounded from 0 random variable λ such that d_N

j

j , λ

= O((log j)^−µ), then 1

log n

n

X

j=1

1 j



f a_j(X_Nj−k:N_j− b_j)
− Z ∞

−∞

f (x)H_G,k,λ(dx)

 a.s.

−→ 0, as n → ∞.

Corollary 2. Let {Xn, n ≥ 1} be a sequence of independent and identically distributed random variables with common distribution function F, and let {N_n, n ≥ 1} be a sequence of pairwise independent random indexes indepen- dent of {X_n, n ≥ 1}. Let us assume (7), (9), and in case when G = G_2,α

with some α > 0, (12) hold. Then

1 log n

n

X

j=1

1 j

 Ih

a_j

1≤i≤Nmaxj

X_i− b_j

≤ xi

− G^Nj/j(x)

 a.s.

−→ 0, as n → ∞.

Let f (.) be a a.e. continuous, bounded real function, such that f (−∞) = 0, f (+∞) = 0. If lim inf ^N_j^j > C > 0, then

1 log n

n

X

j=1

1 j

 f

a_j

1≤i≤Nmaxj

X_i− b_j

− Z ∞

−∞

f (x)G^Nj/j(dx)

 a.s.

−→ 0, as n → ∞.

Additionally, if there exists a positive bounded from 0 random variable λ such that d

_N

j

j , λ



= O((log j)^−µ), then 1

log n

n

X

j=1

1 j

 f

 aj



1≤i≤Nmaxj

Xi− b_j

− Z ∞

−∞

f (x)G^λ(dx)



−→ 0,a.s.

as n → ∞.

Putting in Corollary 1 and 2 the sequence N_j = j, a.s., j ≥ 1, we obtain the main results in [6], [8] and [18].

3. Proofs

Lemma 1. Let x, y ∈ [0, 1], α > 0 be arbitrary numbers.

(i) For y > 0, we have

|x^α− y^α| ≤ α|x − y|^α∧1. (ii) For α ≤ 1, we have

|x^α− y^α| ≤ |x − y|

|y|² . (iii) For arbitrary t ∈ N , 1/t > α, we have

|x(− log x)^t− y(− log y)^t| ≤e

α ∧ t(e/α)^t−1

1 − αt |x^1−αt− y^1−αt|.

Proof of Lemma 1. If α ≤ 1 we consider the functions f (x) = x^α−y^αand g(x) = (x − y)^α in the interval [y, 1]. Now f (y) = 0 = g(y) and inequality

f⁰(x) = α

x^1−α ≤ α

(x − y)^1−α = g⁰(x),

ends the proof of (i) in the case x > y. Case x < y follows by symmetry and case x = y is obvious.

When α ∈ (0, 1) let k be chosen such that ¹

2^k < α ≤ _2k−1¹ . Then

(x^2kα−y^2kα) = (x^α−y^α)(x^α+y^α)(x^2α+y^2α)(x^4α+y^4α) . . . (x^2k−1α+y^2k−1α).

Thus, by the above proved case α > 1, we have

|x − y| ≥ |x^α− y^α|yα(1+2+4+···+2^k−1)= |x^α− y^α|y^α(2k−1)≥ |x^α− y^α|y^2−α, which gives (ii).

For proof of Lemma 1 (iii) we consider the case that x ≥ y > 0 and x(− log x)^t≥ y(− log y)^t, firstly. Let us define two functions

f (x) = x(− log x)^t− y(− log y)^t and g(x) = (e/α)^t

1 − αt(x^1−αt− y^1−αt).

Obviously f (y) = g(y) = 0 and

f⁰(x) = (− log x)^t−1(− log x − t) ≤ (− log x)^t, g⁰(x) = (e/α)^tx^−αt. Now we remark that the maximum of the function −x^αlog x in area (0, +∞) is achieved for x = e^−1/αand is equal e/α, which ends the proof of Lemma 1 (iii) in this case.

When x ≥ y > 0 and x(− log x)^t ≤ y(− log y)^t, then −t ≤ log x ≤ 0 (note that function x(− log x)^t is increasing in the interval (0, e^−t)). Thus, putting f (x) = −x(− log x)^t+ y(− log y)^t, and g(x) = t^(e/α)_1−αt^t−1(x^1−αt − y^1−αt), we have f (y) = g(y) = 0 and f⁰(x) = (− log x)^t−1(log x + t) ≤ t(− log x)^t−1, g⁰(x) = t(e/α)^t−1x^−αt, such that the argumentation similar to

the above ends the proof. 

In the paper [12] (Lemma 7) the following lemma was proved.

Lemma 2.

(a) Let {X_n, n ≥ 1} be a sequence of random variables such that X_n→ 0, a.s., as n → ∞, and for some positive real constant K and every n, |X_n| < K, a.s. Then

1 log n

n

X

j=1

Xj

j

−→ 0,a.s. as n → ∞.

(b) Let {X_n, n ≥ 1} be an arbitrary sequence of random variables such that for some µ ∈ (0, 1), we have d(X_n, 0) = O((log n)^−µ) and

|X_n| < K a.s. for some positive constants K. Then 1

log n

n

X

j=1

Xj

j

−→ 0,a.s. as n → ∞.

(c) For every convergent to zero sequence of real numbers {_n, n ≥ 1}, we have

1 log n

n

X

j=1

_j

j −→ 0, as n → ∞.

Lemma 3. Let {X_n, n ≥ 1} be a sequence of independent and identically distributed random variables such that L(X₁) = F (.). Then for every posi- tive integers j, l, k such that j ∧ l ≥ k, we have

(16) P [Xj−k:j ≤ x] =

k

X

t=0

j t



F^j−t(x)(1 − F (x))^t, and

(17) P [X_j−k:j ≤ x, X_l−k:l≤ y] ≤ P [X_j−k:j ≤ x]P [X_l−k:l≤ y]F (x ∨ y)^−j∧l. Proof of Lemma 3. The evaluation (16) is proved in Lemma A.1 whereas the evaluation (17) is a small generalization of Lemma A.2 ([18], p. 422–

424). From Lemma A.2 in case 1 ≤ k ≤ j ≤ l and x ≥ y and inequality P [Xj−k:j ≤ x] ≤ P [X_j:j ≤ x] = F^j(x), we have

P [Xj−k:j ≤ x, X_l−k:l ≤ y] = P [X_l−k:l≤ y]

≤ P [X_j−k:j ≤ x]P [X_l−k:l ≤ y]F (x ∨ y)^−j.

 Proof of Theorem 1. In the whole proof we will use notation ξ_l,j(k) = I[a_j(X_l−k:l− b_j) ≤ x], k ≥ 0. We have

1 log n

n

X

j=1

1 j

Ia_j(X_Nj−k:N_j− b_j) ≤ x − H_G,k,N^θ

j/j(x)

= 1

log n

n

X

j=1

1 j

∞

X

l=1

(ξl,j(k) − Eξl,j(k))I[Nj = l]

+ 1

log n

n

X

j=1

1 j

∞

X

l=1

(Eξl,j(k) − H_G,k,l/j^θ (x))I[Nj = l]

= V1(n) + V2(n), say.

Step 1. At first we consider the case G(x) > 0.

In order to prove that |V₁(n)|−→ 0, we need some upper estimation on the^a.s.

value cov(ξ_h,j(k), ξ_l,i(k)). However, firstly we evaluate

I_h,j;l,i(k; θ) = P^θ[ ˜X_h−k:h≤ v_j, ˜X_l−k:l≤ v_i]−P^θ[ ˜X_h−k:h≤ v_j]P^θ[ ˜X_l−k:l ≤ v_i].

By Lemma 3 and the fact that F is nondecreasing and v_i∨j ≤ v_i ∨ v_j we have

I_h,j;l,i(k; θ) ≤ |F^−θ(h∧l)(v_j∨i) − 1| ∧ 1.

Now we consider the sequence c_h = h(1 − F (vh)). By (2) and Theorem 1.5.1 in Leadbetter [14] we have for x ∈ R, and h → ∞,

c_h → − log G(x).

Since lim_h→∞1 − F (v_h) = lim_h→∞− log(G(x))

h = 0, then we may choose n_o such that 1 − F (v_h) ≤ ¹₄, for every h ≥ n_o. Thus

I_h,j;l,i(k; θ) ≤     



1 −(i ∨ j)(1 − F (v_(i∨j))) i ∨ j

^−θ(l∧h)

− 1     

∧ 1.

From inequalities e^−2x ≤ 1 − x (valid for 0 ≤ x ≤ ¹₄) and |1 − e^x| ≤ |x|e^|x|

we have for i ∨ j > n_o,

(18)

I_h,j;l,i(k; θ) ≤    

e^θ

2c(i∨j)

i∨j (l∧h)− 1    

∧ 1 ≤ 2θc_(i∨j)

i ∨ j (l ∧ h)e

2θc(i∨j) (i∨j) (l∧h)

∧ 1

≤ 2eθc_(i∨j)(l ∧ h) i ∨ j ∧ 1, and for i ∨ j ≤ n_o,

I_h,j;l,i(k; θ) ≤ 1.

On the other hand, by (8)

P [X_h−k:h≤ x]P [X_l−k:l ≤ y]−P [ ˜X_h−k:h≤ x]^θP [ ˜X_l−k:l ≤ y]^θ ≤ α_h(x)+α_l(y), thus in the case h ≤ l, and v_j ≤ v_i

(19) cov(ξ_h,j(k), ξ_l,i(k)) ≤ α_h,l(v_j, v_i) + I_h,j;l,i(k; θ) + α_h(v_j) + α_l(v_i), whereas in the case h ≤ l, and v_j > vi

(20) cov(ξ_h,j(k), ξ_l,i(k)) ≤ I_h,j;l,i(k; θ) + α_h(v_j) + 2α_l(v_i).

Thus from (9)–(11) and (18)–(20) V ar(V₁(n) log n) = X

{l_j∈N^N}

P [N_j = l_j, j ≥ 1]V ar

n

X

h=1

ξ_lh,h h

!

≤ 2

n

X

h=1 h−1

X

j=1

1

jhEαNj,Nh(vj, vh) + 2

n

X

h=1 h−1

X

j=1

θ

jhE N_j ∧ N_h

h ∧ 1



+ 4

n

X

h=1

1

hEαN_h(v_h) log h + log n + log²no = O (log n)^2−µ
. Now we put n = n(k) = 2^k2/µ and by Chebyshev’s inequality and Borel–

Cantelli lemma, we have

V₁(n(k)) −→ 0, as k → ∞, (21)

with probability one. Furthermore, for n(k) < n < n(k + 1) V1(n) = log n(k)

log n V1(n(k))+ 1 log n

n

X

j=n(k)+1

1 j

∞

X

l=1



ξl,j(k)−Eξl,j(k)



I[Nj = l].

Taking into account (_1+k^k )^2/µV₁(n(k)) ≤ ^{log n(k)}_{log n} V₁(n(k)) ≤ V₁(n(k)), (21) and evaluation

1 log n

n

X

j=n(k)+1

1

j ≤ log^n(k+1)_n(k)

log n(k) = C(k + 1)^2/µ

k^2/µ → 0, as k → ∞, we get

n→∞lim V₁(n) = 0 a.s.

Let us prove V₂(n)−→ 0. We have by Lemma 1^a.s.

|V₂(n)| ≤ 1 log n

n

X

j=1

1 j

∞

X

l=1

|P [X_l−k:l≤ v_j] − P^θ[ ˜X_l−k:l ≤ v_j]|I[Nj = l]

+ 1

log n

n

X

j=1

1 j

∞

X

l=1

|P [ ˜X_l−k:l≤ v_j] − H_G,k,l/n(x)|^θ∧1I[N_j = l]

≤ 1

log n

n

X

j=1

1

jEα_Nj(v_j) + 1 log n

n

X

j=1

1 j

∞

X

l=1

k^θ∧1

0≤t≤kmax 1

t!|F^l(v_j)[− log F^l(v_j)]^t

− G^l/j(x)[− log G^l/j(x)]^t|^θ∧1

I[N_j = l]

≤ 1

log n

n

X

j=1

1

jEα_Nj(v_j) + 1 log n

n

X

j=1

1

jk^θ∧1 max

0≤t≤k

1 t!

|F^j(v_j) − G(x)|^θ∧1 G^2θ∧2(x) . Because from (7) we have F (v_n)ⁿ→ G(x), thus by Lemma 2

|V₂(n)|−→ 0, as n → ∞.^a.s.

Thus (13) is proved in case G(x) > 0.

Step 2. Let us consider the case x such that G(x) = 0.

By the part of Theorem proved above and monotonicity the indicator function for arbitrary δ > 0, x ≤ 0, we have

0 ≤ 1 log n

n

X

j=1

1 j



ξ_Nj,j(k) − H_G,k,N^θ

j/j(x)

= 1

log n

n

X

j=1

1

jξ_Nj,j(k)

≤ 1

log j

n

X

j=1

1

jI[ajXNj−k:Nj+ bj < δ]

≤ H_G,k,δ^θ

o(δ) + 1 log n

n

X

j=1

1 jI[Nj

j < δo] + δ.

Then from the arbitrariness of δ > 0, Lemma 2(b) and (12), we have (13).

Step 3. We show the proof of (14) because (15) runs similarly.

For every  > 0 let us define the partition of real axis Π() = {−∞ =

c_o < c₁ < c₂ < · · · < c_m() = ∞} such that sup_{x,y∈(ci,ci+1)}|f (x) − f (y)| <

/2, i = 0, 1, 2, . . . , m() − 1. Let us define A(x) = si, for x ∈ (c_i, ci+1), i = 0, 1, 2, .., m() − 1, where s_i = sup_{t∈(ci,ci+1)}f (t), i = 0, 1, 2, . . . , m() − 1. For a sufficiently large n, we have

1 log n

n

X

j=1

1

jf aj(XNj−k:Nj− b_j)
≤ 1 log n

n

X

j=1

1

jA aj(XNj−k:Nj− b_j)

=

m()

X

k=1

s_kH_G,k,N^θ

j/j((c_k, c_k+1) +  2

= Z ∞

−∞

A(x)H_G,k,N^θ

j/j(dx) +  2

≤ Z ∞

−∞

f (x)H_G,k,N^θ

j/j(dx) + Z ∞

−∞

|A(x) − f (x)|H_G,k,N^θ

j/j(dx) +  2

≤ Z ∞

−∞

f (x)H_G,k,N^θ

j/j(dx) + .

From the arbitrariness of , we get (14).

Step 4. Now we will prove the second part of Theorem 1.

If 0 < G(x) < 1 then, considering the different cases of limiting laws G1,G2,α, and G3,α, and taking into account inequality

|e^x− e^y| ≤ |x − y|(e^x+ e^y), we always obtain

G

Nj

j (x) − G^λ(x)    

≤    

Nj

j − λ    



|e^−x| ∨ |x^−α| ∨ |x^α|

 G

Nj

j (x) + G^λ(x)



≤ C    

N_j j − λ

    .

On the other hand, by Lemma 1, we have 

 H_G,k,N^θ

j/j(x) − H_G,k,λ^θ (x)  

≤ |H_G,k,N

j/j(x) − H_G,k,λ(x)|^θ∧1

≤ k^θ∧1 max

0≤t≤k

G^Nj/j(x) h

− log G^Nj/j(x) it

− G^λ(x) h

− log G^λ(x) it

θ∧1

≤ C max

0≤t≤k

G^Nj/j(x) − G^λ(x)   

(1−βt)(θ∧1)

≤ Cd(N_j/j, λ)(1−βk)(θ∧1)∧ 1,

for every 0 < β < 1/k, which, by Lemma 2(b), proves Theorem 1. For G2,α(x) = 0 or G3,α(x) = 1 the proof of the second part of Theorem 1 is

obvious. 

4. Examples and applications Example 1.

(a) Let {N_j, j ≥ 1} be a sequence of independent random variables such that N_j ∼ β_j+ γ_jP ois(λ_j) (the uncentred and unnormalized Poisson’s law, P [Nj = βj+kγj] = ^λ

k j

k!e^−λj, k = 0, 1, 2, . . . ) for some sequence of nonnegative numbers {λ_j, γ_j, j ≥ 1} and a sequence of numbers {β_j, j ≥ 1}. If

(22) β_j+ γ_jλ_j = O(j),

then (9) holds. On the other hand, if for some δ_o> 0, µ > 0, we have (23) (jδ_o− β_j)(log j)^µ

γ_jpλ_j ≤ C, then (12) holds.

(b) Let {N_j, j ≥ 1} be a sequence of independent random variables such that N_j ∼ β_j + γjB(nj, pj) (the uncentred and unnormalized Bernouilly’s law, P [N_j = β_j + γ_jk] = ⁿ_k^j
p^k_j(1 − p_j)^nj−k, k = 0, 1, 2, . . . , n_j) for some sequence of nonnegative numbers {n_j, γj, j ≥ 1}, numbers {βj, j ≥ 1} and numbers {p_j, j ≥ 1} such that 0 ≤ p_j ≤ 1, j ≥ 1. If

(24) β_j+ γ_jn_jp_j = O(j),

then (9) holds. On the other hand, if for some δ_o> 0, µ > 0, we have (25) (jδ_o− β_j)(log j)^µpn_jp_j(1 − p_j)

γ_j ≤ C,

then (12) holds.

(c) Let {N_j, j ≥ 1} be a sequence of independent random variables such that N_j ∼ β_j + γ_jU (n_j) (the uncentred and unnormalized uniform law, P [Nj = βj + γjk] = _n¹

j, k = 1, 2, . . . , nj) for some sequence of nonnegative numbers {n_j, γj, j ≥ 1} and sequence of numbers {βj, j ≥ 1}. If

(26) βj+ γj

nj+ 1

2 = O(j),

then (9) holds. On the other hand, if for some δ_o> 0, µ > 0, we have (27) (jδ_o− β_j)(log j)^µ

njγj

≤ C, then (12) holds.

Proof of Example 1 (a). Under such defined sequence {N_j, j ≥ 1} we have EN_j = βj + γjλj, j ≥ 1, and

N

X

k=1 k−1

X

j=1

1

jkEN_j∧ N_k

k ∧ 1



≤

N

X

k=1 k−1

X

j=1

β_j+ γ_jλ_j jk² ≤

N

X

k=1

1

k = O (log N )^2−µ
.

Furthermore, it is easy to check that for arbitrary λ > 0, we have sup

k≥0

 λ^k k!e^−λ



≤ max (

λ^[λ]

[λ]!e^−λ, λ^[λ]+1 ([λ] + 1)!e^−λ

) , and by Stirling formulae we have

sup

k≥0

 λ^k k!e^−λ



≤ C

√ 2πλ. Thus

P N_n n < δ₀



=

(δoj−βj)/γj

X

k=0

λ^k_j

k!e^−λj ≤ Cδ_oj − β_j

γjpλ_j , j ≥ 1,

which ends the proof of point (a). The proof of points (b)–(c) is similar and

will be omitted. 

The different constructions of stationary sequences nonidentically dis- tributed random variables {X_n, n ≥ 1}, satisfying conditions α_j(v_j) → 0 or αj,h(vj, vh) → 0 as j, h → ∞ may be found in [11].

References

[1] Aksomaitis, A., Transfer theorems in a max-scheme, Litovsk. Mat. Sb. 29 (2) (1989), 207–211 (Russian).

[2] Barndorff-Nielsen, O., On the limit distribution of the maximum of a random number of independent random variables, Acta. Math. Acad. Sci. Hungar. 11 (1964), 399–403.

[3] Berkes, I., Cs´aki, E., A universal result in almost sure central limit theory, Stoch.

Proc. Appl. 94 (2001), 105–134.

[4] Brosamler, G. A., An almost everywhere central limit theorem, Math. Proc. Cam- bridge Philos. Soc. 104 (1988), 561–574.

[5] Berman, S. M., Limiting distribution of the maximum in the sequence of dependent random variables, Ann. Math. Statist. 33 (1962), 894–908.

[6] Cheng, S., Peng, L., Qi, Y., Almost sure convergence in extreme value theory, Math.

Nachr. 190 (1998), 43–50.

[7] Fahrner, I., Almost Sure Versions of Weak Limit Theorems, Shaker Verlag, Aachen, 2000.

[8] Fahrner, I., Stadm¨uller, U., On almost sure max-limit theorems, Statist. Probab.

Lett. 37 (1998), 229–236.

[9] Galambos, J., The Asymptotic Theory of Extreme Order Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York–Chichester–

Brisbane, 1978.

[10] H¨ormann, S., An extension of almost sure central limit theory, Statist. Probab. Lett.

76 (2006), 191–202.

[11] Jakubowski, A., Asymptotic Independent Representations for Sums and Order Sta- tistics of Stationary Sequences, NCU Press publications, Toruń, 1991.

[12] Krajka, A., Wasiura, J., On the almost sure central limit theorem for randomly in- dexed sums, Math. Nachr. 282 (4) (2009), 569–580.

[13] Lacey, M. T., Philipp, W., A note on the almost sure central limit theorem, Statist.

Probab. Lett. 9 (1990), 201–205.

[14] Leadbetter, M. R., Lindgren, G., Rootzen, H., Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin, 1983.

[15] O’Brien, G. L., The maximum term of uniformly mixing stationary process, Z. Wahr.

verw. Gebiete 30 (1974), 57–63.

[16] Robbins, H., The asymptotic distribution of the sums of a random number of random variables, Bull. Amer. Math. Soc. 54 (1948), 1151–1161.

[17] Resnick, S. I., Extreme Values. Regular Variation and Point Processes, Springer, 1987.

[18] Stadtm¨uller, U., Almost sure versions of distributional limit theorems for certain order statistics, Statist. Probab. Lett. 58 (2002), 413–426.

[19] Schatte, P., On strong version of the central limit theorem, Math. Nachr. 137 (1988), 249–256.

Andrzej Krajka Zdzisław Rychlik

Maria Curie-Skłodowska University Maria Curie-Skłodowska University Pl. Marii Curie-Skłodowskiej 1 Pl. Marii Curie-Skłodowskiej 1

20-031 Lublin 20-031 Lublin

Poland Poland

e-mail: andrzej.krajka@umcs.lublin.pl e-mail: rychlik@hektor.umcs.lublin.pl Joanna Wasiura-Maślany

The John Paul II Catholic University of Lublin Aleje Racławickie 14

20-950 Lublin Poland

e-mail: jwaaa@wp.pl

Received November 14, 2019