Comptes Rendus

(1)

Comptes Rendus

Mathématique

Lê Vˇan Thành

On the Baum–Katz theorem for sequences of pairwise independent random variables with regularly varying normalizing constants

Volume 358, issue 11-12 (2020), p. 1231-1238.

<https://doi.org/10.5802/crmath.139>

Some rights reserved.

This article is licensed under the

Creative Commons Attribution4.0International License.

http://creativecommons.org/licenses/by/4.0/

LesComptes Rendus. Mathématiquesont membres du Centre Mersenne pour l’édition scientifique ouverte

www.centre-mersenne.org

(2)

2020, 358, n11-12, p. 1231-1238

https://doi.org/10.5802/crmath.139

Probability Theory /Probabilités

On the Baum–Katz theorem for sequences of pairwise independent random variables with regularly varying normalizing constants

Lê Vˇan Thành^a

aDepartment of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam E-mail:[email protected]

Abstract.This paper proves the Baum–Katz theorem for sequences of pairwise independent identically distributed random variables with general norming constants under optimal moment conditions. The proof exploits some properties of slowly varying functions and the de Bruijn conjugates, and uses the techniques developed by Rio (1995) to avoid using the maximal type inequalities.

2020 Mathematics Subject Classification.60F15.

Manuscript received 5th June 2020, revised 5th November 2020, accepted 7th November 2020.

1. Introduction and result

Let 1≤ p <2,αp ≥1 and {X,Xn,n ≥1} be a sequence of pairwise independent identically distributed (p.i.i.d.) random variables. In this paper, by using some results related to slowly varying functions and techniques developed by Rio [10], we provide the necessary and sufficient conditions for

X

n

n^α^p⁻²P Ã

max

1≤k≤n

¯

k

X

i=1

X_i

¯

>εn^1/^αeL(n^1/^α)

!

< ∞ for allε>0, (1) where L(e·) is the de Bruijn conjugate of a slowly varying function L(·). The result provides the rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers (SLLN) with regularly varying normalizing constants. When the random variables are i.i.d. withE(X)

=0,E(|X|^p)< ∞, andL(·)=1, (1) was obtained by Baum and Katz [2].

The notion of regularly varying function can be found in Seneta [12, Chapter 1]. A real-valued functionR(·) is said to be regularly varying with index of regular variationρ∈Rif it is a positive and measurable function on [A,∞) for someA>0, and for eachλ>0,

xlim→∞

R(λx) R(x) =λ^ρ.

(3)

1232 Lê Vˇan Thành

A regularly varying function with the index of regular variationρ=0 is called to be slowly varying.

It is well known that a functionR(·) is regularly varying with the index of regular variationρif and only if it can be written in the form

R(x)=x^ρL(x)

whereL(·) is a slowly varying function (see, e.g., Seneta [12, p. 2]). Seneta [11] (see also in Bingham et al. [3, Lemma 1.3.2]) proved that ifL(·) is a slowly varying function defined on [A,∞) for someA>0, then there exists B≥Asuch thatL(x) is bounded on every finite closed interval [a,b]⊂[B,∞). Galambos and Seneta [7, p. 111] showed that for any slowly varying functionL(x), there exists a differentiable slowly varying functionL₁(·) defined on [B,∞) for someB≥Asuch that

x→∞lim L(x)

L₁(x)=1 and lim

x→∞

xL⁰₁(x) L₁(x) =0.

Conversely, ifL(·) is a positive differentiable function satisfying

x→∞lim xL⁰(x)

L(x) =0, (2)

thenL(·) is a slowly varying function. IfL(·) be a differentiable slowly varying function satisfying (2), then by direct calculations (see, e.g. [1, Lemma 2.3]), we can show that for allp>0, there existsB>0 such that

x^pL(x) is strictly increasing on [B,∞), x⁻^pL(x) is strictly decreasing on [B,∞). (3) LetL(·) be a slowly varying function. Then by [3, Theorem 1.5.13], there exists a slowly varying functionL(e·), unique up to asymptotic equivalence, satisfying

x→∞limL(x)eL(xL(x))=1 and lim

x→∞L(x)Le ¡ xeL(x)¢

=1. (4)

The functionLeis called the de Bruijn conjugate of L, and (L,L) is called a (slowly varying)e conjugate pair (see, e.g., Bingham et al. [3, p. 29]). By [3, Proposition 1.5.14], if (L,L) is a conjugatee pair, then for a,b,α >0, each of (L(ax),L(bx)), (aL(x),e a⁻¹L(x)), ((L(xe ^α))^1/α, (eL(x^α))^1/α) is a conjugate pair. Bojani´c and Seneta [4] (see also [3, Theorem 2.3.3 and Corollary 2.3.4] in Bingham et al.) proved that ifL(·) is a slowly varying function satisfying

x→∞lim

µL(λ0x) L(x) −1

¶

log(L(x))=0, (5)

for someλ0>1, then for allα∈R,

x→∞lim

L(xL^α(x))

L(x) =1, (6)

and therefore, we can choose (up to aymptotic equivalence)eL(x)=1/L(x). Especially, if for some γ∈R,L(x)=log^γ(x+2),x≥0, thenL(x)e =1/L(x). For α,β>0 and for f(x)=x^β/αL^1/α(x^β), g(x)=x^α^/^βeL^β(x^α), we have (see [3, Theorem 1.5.12 and Proposition 1.5.15])

xlim→∞

f(g(x)) x = lim

x→∞

g(f(x))

x =1. (7)

Here and thereafter, for a slowly varying functionL(·), we denote the de Bruijn conjugate of L(·) byL(e·). We will assume, without loss of generality, thatL(x) andeL(x) are both defined on [0,∞) and differentiable on [A,∞) for someA>0.Theorem 1 is the main result of this paper. The Marcinkiewicz–Zygmund SLLN with regularly varying normalizing constants was also studied recently by Anh et al. [1], where the proof is based on the Kolmogorov maximal inequality.

Theorem 1. Let1≤p<2, and let{X,X_n,n≥1}be a sequence of p. i.i.d. random variables, L(·) a slowly varying function defined on[0,∞). When p=1, we assume further that L(x)≥1and is increasing on[0,∞). Then the following statements are equivalent.

C. R. Mathématique,2020, 358, n11-12, 1231-1238

(4)

(i) The random variable X satisfies E(X)=0,E¡

|X|^pL^p(|X|)¢

< ∞. (8)

(ii) For allα≥1/p, we have X∞

n=1

n^α^p⁻²P Ã

1max≤j≤n

¯

j

X

i=1

X_i

¯

>εn^αLe¡ n^α¢

!

< ∞for allε>0. (9) (iii) The Marcinkiewicz–Zygmund-type SLLN

n→∞lim

max₁_≤_k_≤_n¯

¯Pk i=1X_i¯

¯ n^1/pLe¡

n^1/p¢ =0 almost surely (a.s.) (10) holds.

For a sequence of random variables which are pairwise independent but not identically distributed, Csörg˝o et al. [5] proved that the Kolmogorov condition alone does not ensure the SLLN. On the case where the random variables {X,X_n,n≥1} are p. i.i.d, Etemadi [6] proved that the Kolmogorov SLLN holds under moment conditionE(|X|)< ∞. For γ>0, Martikainen [9]

proved that

nlim→∞

Pn i=1Xi

nlog^−γ(n)=0 a.s.

if and only ifE(X)=0 andE(|X|log^γ(|X| +2))< ∞. This is a special case of Theorem 1 whenp=1 andL(x)≡log^γ(x+2). For the case where 1<p<2, Martikainen [8] proved that ifE(X)=0 and E(|X|^plog^r(|X| +1))< ∞for somer >max{0, 4p−6}, then the Marcinkiewicz–Zygmund SLLN holds. By lettingL(x)≡1, we obtain the following corollary. Whenα=1/p, this corollary was obtained by Rio [10].

Corollary 2. Let1≤p<2, and let{X,Xn,n≥1}be a sequence of p. i.i.d. random variables. Then the following statements are equivalent.

(i) The random variable X satisfies

E(X)=0,E¡

|X|^p¢

< ∞. (ii) For allα≥1/p, we have

X∞ n=1

n^α^p⁻²P Ã

1max≤j≤n

¯

j

X

i=1

X_i

¯

>εn^α

!

< ∞for allε>0.

(iii) The Marcinkiewicz–Zygmund SLLN

n→∞lim

max₁_≤_k_≤_n¯

¯Pk i=1X_i¯

¯ n^1/p =0 a.s.

holds.

2. Proof

To prove the main result, we firstly introduce some preliminaries. Through this paper,C(·), C₁(·),C₂(·), . . . denote constants which depend only on variables appearing in the parentheses.

Lemma 3 is a direct consequence of Karamata’s theorem (see [3]).

Lemma 3. Let a,b>1, and L(·)be a differentiable slowly varying function defined on[0,∞). Then

n

X

k=1

a^kL³ b^k´

≤C₁(a,b)aⁿL¡ bⁿ¢

.

(5)

Lemma 4 gives simple criterions forE(|X|^pL^p(|X|))< ∞. Whenα=1/p, the equivalence of (11) and (12) was established by Anh et al. [1, Proposition 2.6].

Lemma 4. Let p≥1,αp≥1, and X be a random variable. Let L(x)be a slowly varying function defined on[0,∞), and b_n =n^αLe(n^α), n≥1. Assume that x^1/^αL^1/^α(x)and x^αL(xe ^α)are strictly increasing on[A,∞)for some A>0. Then the following statements are equivalent.

E¡

|X|^pL^p(|X|)¢

< ∞. (11)

X∞ n=1

n^αp−1P(|X| >b_n)< ∞. (12)

X∞ n=1

2ⁿ^α^pP¡

b₂n−1< |X| ≤b2ⁿ¢

< ∞. (13)

Proof. Letf(x)=x^1/^αL^1/^α(x),g(x)=x^αeL(x^α). By using (7) withβ=1, we have f¡

g(x)¢

∼g¡ f(x)¢

∼xasx→ ∞. (14)

Firstly, we will prove (11) is equivalent to (12). For a non negative random variableY andr>0, EY^r< ∞if and only ifP_∞

n=1n^r⁻¹P(Y >n)< ∞. Applying this, we have thatE(f^α^p(|X|))< ∞if and only if

X∞ n=1

n^α^p⁻¹P¡

f(|X|)>n¢

< ∞. (15)

Combining (14) with the assumption thatf(x) andg(x) are strictly increasing on [A,∞), we see that (15) is equivalent to

X∞ n=1

n^α^p⁻¹P¡

|X| >n^αLe¡ n^α¢¢

< ∞.

The proof of the equivalence of (11) and (12) is completed. Now, we will prove (11) is equivalent to (13). Fornlarge enough, on event (b₂n−1< |X| ≤b₂n), we have

f^αp¡ b₂n−1

¢<f^αp(|X|)≤f^αp(b₂ⁿ) , or equivalently,

¡f¡ g¡

2ⁿ⁻¹¢¢¢αp

< |X|^pL^p(|X|)≤¡ f¡

g¡ 2ⁿ¢¢¢_αp

. (16)

Combining (14) and (16), we see that (13) is equivalent to (11).

Proof of Theorem 1. By the arguments leading to (2) and (3), without loss of generality, we can assume that there exists a positive integerAlarge enough such thatx^1/^αL(x^1/^α), x^αL(xe ^α) and x^p⁻¹L^p(x) (forp>1) are strictly increasing on [A,∞).

Firstly, we prove the implication ((i)⇒(ii)). It is easy to see that (9) is equivalent to X∞

n=1

2ⁿ⁽^α^p⁻¹⁾P Ã

1≤maxj<2ⁿ

¯

j

X

i=1

X_i

¯

>ε2ⁿ^αeL¡ 2ⁿ^α¢

!

< ∞for allε>0. (17) Assume that (8) holds. Forn≥1, setb_n=n^αeL(n^α) and

X_i_,n=X_i1(|X_i| ≤b_n) , 1≤i≤n.

For allε>0 andn≥1, we have P

Ã max

1≤j<2ⁿ

¯

j

X

i=1

X_i

¯

>εb₂ⁿ

!

≤P µ

max

1≤i<2ⁿ|X_i| >b₂ⁿ

¶ +P

Ã max

1≤j<2ⁿ

¯

j

X

i=1

X_i_{, 2}ⁿ

¯

>εb₂ⁿ

!

≤P µ

1max≤i<2ⁿ|X_i| >b₂ⁿ

¶ +P

Ã

1max≤j<2ⁿ

¯

j

X

i=1

(X_i_{, 2}ⁿ−EX_i_{, 2}ⁿ)

¯

>εb₂ⁿ−

2ⁿ

X

i=1

¯¯E(X_i_{, 2}ⁿ)¯

¯

! . (18)

(6)

SinceL(e·) is slowly varying,b₂n+1≤4^αb₂ⁿforn≥n₀for somen₀≥A. By the second half of (8) and Lemma 4, we have

∞ >

X∞ j=1

j^α^p−1P¡

4^α|X| >bj¢

= X∞ j=1

j^α^p−2

j

X

i=1

P¡

4^α|Xi| >bj¢

≥ X∞ j=1

j^α^p⁻²P µ

1max≤i≤j4^α|X_i| >b_j

¶

= X∞ n=0

2ⁿ⁺¹−1

X

j=2ⁿ

j^α^p⁻²P µ

1max≤i≤j4^α|X_i| >b_j

¶

≥1 2

X∞ n=0

2ⁿ⁺¹−1

X

j=2ⁿ

2ⁿ⁽^α^p⁻²⁾P µ

1max≤i<2ⁿ4^α|X_i| >b_j

¶

≥1 2

X∞ n=n0

2^n(αp−1)P µ

max

1≤i<2ⁿ|X_i| >b₂ⁿ

¶ .

(19)

Forn≥1, the first half of (8) imply that

¯

n

X

i=1

E¡ X_i,n¢

¯

b_n ≤

n

X

i=1

¯

¯EX_i1 µ

|X_i| >b_n

¶ ¯

¯

¯ b_n

≤nE(|X|1(|X| >b_n))

b_n .

(20)

Fornlarge enough and forω∈(|X| >b_n), we have n

b_n ≤n^(p−1)^αLe^p−1(n^α) Le^p(n^α) =

¡n^αeL(n^α)¢p−1

L^p¡

n^αL(ne ^α)¢ Le^p(n^α)L^p¡

n^αL(ne ^α)¢

≤2b^p−1_n L^p(b_n)≤2|X(ω)|^p−1L^p(|X(ω)|),

(21)

where we have applied the second half of (4) in the second inequality and the monotonicity of x^p−1L^p(x) in the third inequality. It follows from (21) and the second half of (8) that

n|E(|X|1(|X| >b_n))| b_n ≤2E¡

|X|^pL^p(|X|)1(|X| >bn)¢

→0 asn→ ∞. (22)

From (18), (19), (20) and (22), the proof of (17) will be completed if we can show that X∞

n=1

2ⁿ⁽^α^p⁻¹⁾P Ã

1≤maxj<2ⁿ

¯

j

X

i=1

¡X_i_{, 2}ⁿ−EX_{i, 2}ⁿ¢

¯

≥εb₂n−1

!

< ∞for allε>0. (23) Form≥0, setS0,m=0 and

S_j,m=

j

X

i=1

¡X_i_{, 2}^m−EX_i_{, 2}^m¢ , j≥1.

Now, we use techniques developed by Rio [10]. For 1≤j <2ⁿ and for 0≤m≤n, letk= bj/2^mc be the greatest integer which is less than or equal to j/2^m. Then 0≤k<2^n−mandk2^m≤j <

(k+1)2^m. Letj_m=k2^m, then S_j,n=

n

X

m=1

¡S_j_m₋₁_,m−1−S_j_m_,m−1¢ +

n

X

m=1

¡S_j,m−S_j_,m−1−S_j_m_,m+S_j_m_,m−1¢

, 1≤j<2ⁿ. (24) and

¯¯S_j,m−S_j,m−1−S_j_m_,m+S_j_m_,m−1¯

¯≤

jm+2^m

X

i=jm+1

¡¯¯X_{i, 2}^m−X_{i, 2}m−1

¯

¯+E¯

¯X_i_{, 2}^m−X_{i, 2}m−1

¯

¢. (25) Set

Yi,m=¯

¯Xi, 2^m−X_i_{, 2}m−1

¯

¯−E¡¯

¯Xi, 2^m−X_{i, 2}m−1

¯

¢,m≥1,i≥1.

(7)

It follows from (25) that

¯¯S_j,m−S_j_,m−1−S_j_m_,m+S_j_m_,m−1¯

¯≤

jm+2^m

X

i=jm+1

Y_i,m+2^m⁺¹E¡¯

¯X_{1, 2}^m−X_{1, 2}m−1¯

¯

¢. (26)

By the definition ofj_m, we have either j_m−1=j_morj_m−1=j_m+2^m−1. Therefore

¯¯S_j_m−1_,m−1−S_j_m_,m−1¯

¯≤

¯

jm+2^m⁻¹

X

i=jm+1

¡X_i_{, 2}m−1−E(X_i_{, 2}m−1)¢

¯

. (27)

Combining (24), (26) and (27), we have

1max≤j<2ⁿ

¯¯S_j,n¯

¯≤

n

X

m=1

max

0≤k<2^n−m

¯

k2^m+2^m−1

X

i=k2^m+1

¡X_{i, 2}m−1−E¡

X_{i, 2}m−1¢¢

¯

¯ +

n

X

m=1

max

0≤k<2^n−m

¯

(k+1)2^m

X

i=k2^m+1

Y_i_,m

¯

¯ +

n

X

m=1

2^m⁺¹E¡¯

¯X_{1, 2}^m−X_{1, 2}m−1¯

¯

¢. (28)

It follows from (22) that

2^mE¡

|X|1¡

|X| >b₂m−1¢¢

b₂^m →0 as m→ ∞. (29)

From (29) and Lemma 3, we can apply Toeplitz’s lemma and conclude that

nlim→∞

n

X

m=1

2^mE µ

|X|1 µ

|X| >b₂m−1

¶¶

b₂ⁿ =0. (30)

By using

E¡¯

¯X_{1, 2}^m−X_{1, 2}m−1

¯

¢≤E¡

|X|1¡

|X| >b₂m−1¢¢

,m≥A, we have from (30) that

nlim→∞

n

X

m=1

2^m+1E µ¯

¯

X_{1, 2}^m−X_{1, 2}m−1

¯

¶

b2ⁿ =0. (31)

Letε1>0 be arbitrary, and letaandbbe positive constants satisfying αp/2<a<α,a+b=α.

Forn≥1, 0≤m≤n, letλm,n=ε12^bm2^anL(2e ⁿ^α). Then

n

X

m=1

λm,n=ε12^anLe¡ 2^nα¢ Xⁿ

m=1

2^mb

=ε12^an2^bLe¡

2ⁿ^α¢2^bn−1

2^b−1 ≤2^bε1b2ⁿ

2^b−1 :=C1(b)ε1b2ⁿ.

(32)

(8)

By (32) and Chebyshev’s inequality, we have

P Ã n

X

m=1

max

0≤k<2ⁿ⁻^m

¯

(k+1)2^m

X

i=k2^m+1

Y_i_,m

¯

≥C₁(b)ε1b₂ⁿ

!

≤

n

X

m=1

P Ã

0≤maxk<2ⁿ⁻^m

¯

(k+1)2^m

X

i=k2^m+1

Y_i_,m

¯

≥λm,n

!

≤

n

X

m=1

λ⁻²m,nE Ã

max

0≤k<2ⁿ⁻^m

¯

(k+1)2^m

X

i=k2^m+1

Y_i_,m

¯

!2

≤

n

X

m=1

λ⁻m,n² 2^n−m−1

X

k=0

E Ã_(k

+1)2^m

X

i=k2^m+1

Y_i,m

!2

≤

n

X

m=1

2ⁿλ⁻m,n² E³

¯¯X_i_{, 2}^m−X_i_{, 2}m−1¯

¯

2´

≤

n

X

m=1

2ⁿ⁺¹λ⁻²m,n

³E³ X_{i, 2}² m

´+E³ X_i²_{, 2}m−1

´´

=

n

X

m=1

2ⁿ⁺¹λ⁻²m,n

¡E¡

X²1(|X| ≤b₂^m)¢ +E¡

X²1¡

|X| ≤b₂m−1¢¢¢

, (33)

and

P Ã _n

X

m=1

max

0≤k<2^n−m

¯

k2^m+2^m−1

X

i=k2^m+1

¡X_i_{, 2}m−1−E¡

X_i_{, 2}m−1¢¢

¯

≥C₁(b)ε1b₂ⁿ

!

≤

n

X

m=1

P Ã

max

0≤k<2ⁿ⁻^m

¯

k2^m+2^m−1

X

i=k2^m+1

¡X_{i, 2}m−1−E¡ X_i_{, 2}m−1

¢¢

¯

≥λm,n

!

≤

n

X

m=1

λ⁻²m,nE Ã

0≤kmax<2ⁿ⁻^m

¯

k2^m+2^m⁻¹

X

i=k2^m+1

¡X_i_{, 2}m−1−E¡

X_{i, 2}m−1¢¢

¯

!2

≤

n

X

m=1

λ⁻²m,n 2ⁿ⁻^m−1

X

k=0

E

Ãk2^m+2^m⁻¹

X

i=k2^m+1

¡X_i_{, 2}m−1−E¡

X_i_{, 2}m−1¢¢

!2

≤

n

X

m=1

2ⁿλ⁻²m,nE³ X_i²_{, 2}m−1

´

=

n

X

m=1

2ⁿλ⁻²m,nE¡ X²1¡

|X| ≤b₂m−1¢¢

. (34)

Sinceαp<2aand 1≤p<2, elementary calculations show that X∞

n=1

2^n(α^p⁻¹⁾

n

X

m=1

2ⁿλ⁻²m,n≤ 1 ε²₁(4^b−1)

X∞ n=1

2^n(α^p^−2a)Le⁻²¡ 2ⁿ^α¢

< ∞. (35)

We recall thatb_nis strictly increasing on [A,∞). From (28), (31), and (33)–(35), the proof of (23) is completed if we can show that

I:= X∞ n=A

2ⁿ⁽^α^p−1)

n

X

m=A

2ⁿλ⁻²m,n m

X

k=A

b²

2^kP¡

b₂k−1< |X| ≤b₂k¢

< ∞. (36)

(9)

By using the second half of (8), Lemmas 3–4, and definition ofaandb, we have I=

X∞ n=A

2ⁿ(αp−1)ε⁻₁²2ⁿ⁽¹⁻^2a)Le⁻²¡ 2ⁿ^α¢

Ã _n X

m=A

2⁻^2mb

m

X

k=A

b²

2^kP¡

b₂k−1< |X| ≤b₂k

¢

!

≤C₂(b)ε⁻₁² X^∞

n=A

2ⁿ⁽^α^p⁻^2a)Le⁻²¡ 2ⁿ^α¢

n

X

k=A

2⁻^2kbb²

2^kP¡

b₂k−1< |X| ≤b₂k

¢

≤C₂(b)ε⁻²₁ X^∞

k=A

Ã∞

X

n=k

2ⁿ(αp−2a) Le⁻²¡

2^nα¢

! 2^−2kbb²

2^kP¡

b₂k−1< |X| ≤b₂k

¢

≤C(α,a,b,p)ε⁻1²

X∞ k=A

2^k(αp−2a)

Le⁻²(2^k^α)2⁻^2kbb²

2^kP¡

b₂k−1< |X| ≤b₂k

¢

=C(α,a,b,p)ε⁻1²

X∞ k=A

2^k^α^pP¡

b₂k−1< |X| ≤b₂k

¢< ∞ thereby proving (36). The proof of the implication ((i)⇒(ii)) is completed.

By choosingα=1/p, we have the implication ((ii)⇒(iii)). The proof of the implication ((iii)

⇒(i)) follows from the Borel–Cantelli lemma for pairwise independent events and Lemma 4

(see [1, the proof of Theorem 3.1]).

References

[1] V. T. N. Anh, N. T. T. Hien, L. V. Thành, V. T. H. Van, “The Marcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences”,J. Theor. Probab.(2019), p. 1-18.

[2] L. E. Baum, M. Katz, “Convergence rates in the law of large numbers”,Trans. Am. Math. Soc.120(1965), no. 1, p. 108- 123.

[3] N. H. Bingham, C. M. Goldie, J. L. Teugels,Regular variation, paperback ed. ed., Encyclopedia of Mathematics and Its Applications, vol. 27, Cambridge University Press, 1989.

[4] R. Bojanic, E. Seneta, “Slowly varying functions and asymptotic relations”,J. Math. Anal. Appl.34(1971), no. 2, p. 302- 315.

[5] S. Csörg˝o, K. Tandori, V. Totik, “On the strong law of large numbers for pairwise independent random variables”,Acta Math. Hung.42(1983), no. 3-4, p. 319-330.

[6] N. Etemadi, “An elementary proof of the strong law of large numbers”,Z. Wahrscheinlichkeitstheor. Verw. Geb.55 (1981), no. 1, p. 119-122.

[7] J. Galambos, E. Seneta, “Regularly varying sequences”,Proc. Am. Math. Soc.41(1973), no. 1, p. 110-116.

[8] A. Martikainen, “On the strong law of large numbers for sums of pairwise independent random variables”,Stat.

Probab. Lett.25(1995), no. 1, p. 21-26.

[9] A. I. Martika˘ınen, “A remark on the strong law of large numbers for sums of pairwise independent random variables”, J. Math. Sci.75(1995), no. 5, p. 1944-1946.

[10] E. Rio, “Vitesses de convergence dans la loi forte pour des suites dépendantes (Rates of convergence in the strong law for dependent sequences)”,C. R. Acad. Sci. Paris320(1995), no. 4, p. 469-474.

[11] E. Seneta, “An interpretation of some aspects of Karamata’s theory of regular variation”,Publ. Inst. Math.15(1973), no. 29, p. 111-119.

[12] ——— ,Regularly varying functions, Lecture Notes in Mathematics, vol. 508, Springer, 1976.