Statistics & Probability Letters 64 (2003) 121–131

A class of strong limit theorems for the sequences ofarbitrary random variables

Wen Liua ;∗, Weiguo Yangb

aDepartment of Applied Mathematics, Hebei University of Technology, Tianjin 300130, ChinabFaculty of Science, Jiangsu University, Zhenjiang 212013, China

Received February 2002; received in revised form February 2003

Abstract

In this paper, the strong limit theorems for arbitrary stochastic sequences are studied. Some convergencetheorems for martingale di,erence sequence and a class of strong limit theorem for countable nonhomogeneousMarkov chains are the particular cases of the results of this paper. Finally, the strong law of large numbersfor the *-mixing sequence and the strong law of large numbers for the limit martingale are proved.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Limit theorems; Martingale di,erence sequence; Markov chains

1. Introduction

Let {Xn;Fn; n¿ 0} be a stochastic sequence on the probability space (�;F; P), that is, the se-quence of �-6elds {Fn; n¿ 0} in F is increasing in n (that is Fn ↑), and {Fn} are adapted torandom variables {Xn}. The purpose of this paper is to study the strong limit theorem for arbitrarystochastic sequences. As corollaries, some convergence theorems for martingale di,erence sequence,a class of strong limit theorems for Markov process are obtained. Chung’s classical strong law oflarge numbers for the sequence of independent random variables and Chow’s convergence theoremsfor martingale di,erence sequence and Liu and Liu’s strong limit theorem for countable nonhomo-geneous Markov chains are particular cases of the results of this paper. Finally, the strong law oflarge numbers for the *-mixing sequence and the strong law of large numbers for limit martingaleare proved. In the proof, a new approach in the study of a.e. convergence used in the works (Liu,1990; Liu and Liu, 1994) is applied. The essence of the approach is to prove the existence of a.e.convergence at certain random sequence by using Doob’s martingale convergence theorem.

∗ Corresponding author. Tel.: +86-22-27357488; fax: +86-22-27355757.E-mail address: anniemanhong@sina.com (W. Liu).

0167-7152/03/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0167-7152(03)00104-4

122 W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131

2. Main results

Theorem 1. Let {Xn;Fn; n¿ 0} be a stochastic sequence de8ned as before, and let {an; n¿ 0} bean increasing sequence of positive numbers. Let {’n(x); n¿ 0} be a sequence of nonnegative, evenfunctions on R such that as |x| increases

’n(x)=|x| ↑; ’n(x)=x2 ↓ : (1)

Let

A=

{! :

∞∑n=1

E[’n(Xn)|Fn−1]=’n(an)¡+ ∞}: (2)

Then

∞∑n=1

(Xn − E[Xn |Fn−1])=an converges a:e: on A: (3)

If an ↑ ∞, then

limn→∞

1an

n∑m=1

{Xm − E[Xm |Fm−1]} = 0 a:e: on A: (4)

Proof. Let n¿ 0, X ∗n =XnI(|Xn|6 an). Let k be a positive integer number, and let Zn=’n(Xn)=’n(an),

Ak =

{! :

∞∑n=1

E[Zn |Fn−1]6 k

}; (5)

�k =min

{n : n¿ 1;

n+1∑i=1

E[Zi |Fi−1]¿k

}; (6)

where �k = +∞, if the right-hand side of (6) is empty, then∑�k

n=1 Zn =∑∞

n=1 I(�k¿ n)Zn. SinceI(�k¿ n) is measurable Fn−1, and Zn is nonnegative, we have

E

(�k∑n=1

Zn

)=E

( ∞∑n=1

I(�k¿ n)Zn

)

=E

{ ∞∑n=1

E[I(�k¿ n)Zn|Fn−1]

}

W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131 123

=E

{ ∞∑n=1

I(�k¿ n)E[Zn|Fn−1]

}

=E

{�k∑n=1

E[Zn|Fn−1]

}6 k: (7)

Since Ak = {�k =+∞}, we have by (7)∞∑n=1

∫Ak

Zn dP =∞∑n=1

E(I(Ak)Zn) = E

{I(Ak)

∞∑n=1

Zn

}

= E

{I(�k =+∞)

∞∑n=1

Zn

}

= E

{I(�k =+∞)

�k∑n=1

Zn

}

6E

(�k∑n=1

Zn

)6 k: (8)

By (1), ’n(x) ↑ as |x| increases. We have by (8)∞∑n=1

P(Ak(X ∗n = Xn)) =

∞∑n=1

∫Ak (|xn|¿an)

dP

6∞∑n=1

∫Ak (|xn|¿an)

’n(Xn)’n(an)

dP

6∞∑n=1

∫Ak

Zn dP6 k:

By Borel–Cantelli lemma, we have P(Ak(X ∗n = Xn); i:o:) = 0. Hence we have

∞∑n=1

(X ∗n − Xn)=an converges a:e: on Ak: (9)

Since A=⋃

k Ak , it follows from (9) that∞∑n=1

(X ∗n − Xn)=an converges a:e: on A: (10)

Let

Ym = (X ∗m − E[X ∗

m |Fm−1])=am; m¿ 1:

124 W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131

Let = 1;−1, and let

tn( ) =exp{ ∑n

m−1 Ym}∏nm=1 E[exp{ Ym}|Fm−1]

; n¿ 1: (11)

It is easy to see that the sequence {tn( ); n¿ 1} is a martingale. Since E|tn( )|=Etn( )=Et1( )=1,by Doob’s martingale convergence theorem, we have

limn→∞ tn( ) exists and is 6nite a:e: (12)

By inequality

06 ex − 1 − x6 x2e|x| ∀x∈R;

and noticing that |Yn|6 2; E[Yn|Fn−1] = 0 a.e., and

E[Y 2n |Fn−1] = E[((X ∗

n − E[X ∗n |Fn−1])=an)2 |Fn−1]

= (E[(X ∗n )

2 |Fn−1] − (E[X ∗n |Fn−1])2)=a2n

6E[(X ∗n )

2 |Fn−1]=a2n a:e:;

we have

06E[exp{ Yn} |Fn−1] − 1 = E[exp{ Yn} − Yn − 1) |Fn−1]

6E[ 2Y 2n e

| Yn| |Fn−1]6 e2E[Y 2n |Fn−1]

6 e2E[(X ∗n )

2 |Fn−1]=a2n a:e: (13)

By (1), x2=a2n6’n(x)=’n(an), as |x| = an. Hence we have

(X ∗n )

2

a2n6

’n(X ∗n )

’n(an)6

’n(Xn)’n(an)

: (14)

By (13) and (14)

06E[exp{ Yn} |Fn−1] − 16 e2E[’n(Xn) |Fn−1]

’n(an)a:e: (15)

It follows from (15) and (2) that∞∑n=1

(E[exp{ Yn} |Fn−1] − 1) converges a:e: on A: (16)

Or, equivalently∞∏n=1

E[exp{ Yn} |Fn−1] converges a:e: on A: (17)

W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131 125

By (11), (12) and (17) we have

limn→∞ exp

{

n∑m=1

Ym

}exists and is 6nite a:e: on A: (18)

Since (18) holds for = 1, and −1, we have∞∑n=1

Yn =∞∑n=1

(X ∗n − E[X ∗

n |Fn−1])=an converges a:e: on A: (19)

By (1), |x|=an6’n(x)=’n(an), as |x|¿an. Hence

|(E[Xn |Fn−1] − E[X ∗n |Fn−1])=an| = |E[(Xn − X ∗

n )=an |Fn−1]|

6E[|Xn − X ∗n |=an |Fn−1]

= E[(|Xn|=an)I(|Xn|¿an) |Fn−1]

6E[’n(Xn)=’n(an))I(|Xn|¿an) |Fn−1]

6E[’n(Xn) |Fn−1]=’n(an) a:e: (20)

By (20) and (2), we have∞∑n=1

(E[Xn |Fn−1] − E[X ∗n |Fn−1])=an converges a:e: on A: (21)

By (10), (19) and (21), (3) follows.If moreover an ↑ ∞, (4) follows from (3) and the Kronecker Lemma.

Corollary 1. Let {Xn; n¿ 0} be a sequence of arbitrary random variables. Let Fn = �(X0; : : : ; Xn) and F−n = {�; ∅}; n¿ 1. Let ’n and an de8ned as before. If

∞∑n=1

E[’n(Xn)]’n(an)

¡+ ∞; (22)

then for any m¿ 1∞∑n=1

{Xn − E[Xn |Fn−m]}=an converges a:e: (23)

If moreover an ↑ ∞, then for any m¿ 1,

limn→∞

1an

n∑k=1

{Xk − E[Xk |Fk−m]} = 0 a:e: (24)

Proof. By (22) and nonnegativeness of ’n, we have∞∑n=1

E[’n(Xn) |Fn−1]=’n(an)¡∞ a:e:; (25)

which implies P(A) = 1. By (25) and Theorem 1, (23) holds for m= 1.

126 W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131

For m¿ 1, by (22) we have

∞∑n=1

E[’nm+k(Xnm+k)]’nm+k(anm+k)

¡+ ∞; k = 0; 1; 2; : : : ; m − 1:

By nonnegativeness of ’n, for k = 0; 1; 2; : : : ; m − 1

∞∑n=1

E[’nm+k(Xnm+k) |F(n−1)m+k]’nm+k(anm+k)

¡+ ∞ a:e: (26)

Since {Xnm+k ;Fnm+k ; n¿ 0} is a stochastic sequence, by (26) and Theorem 1, we have for k =0; 1; 2; : : : ; m − 1,

∞∑n=1

Xnm+k − E[Xnm+k |F(n−1)m+k]’nm+k(anm+1)

converges a:e: (27)

By (27)

∞∑n=m

Xn − E[Xn |Fn−m]an

=m−1∑k=0

∞∑n=1

Xnm+k − E[Xnm+k |F(n−1)m+k]anm+k

converges a:e:;

namely, (23) holds for m¿ 1. If moreover an ↑ ∞, then (24) holds for m¿ 1.

Corollary 2 (Chung): Let {Xn; n¿ 0} be a sequence of independent random variables, ’n, and ande8ned as before. If (22) holds, then

∞∑n=1

1an

(Xn − E(Xn)) converges a:e: (28)

If moreover an ↑ ∞, then

limn→∞

1an

n∑k=1

(Xk − E(Xk)) = 0 a:e: (29)

This is the classical strong law of large numbers for the sequence of independent random variables(see Chung, 1974, p. 124; Petrov, 1975, p. 267).

Corollary 3. Let {Xn; n¿ 0} be a sequence of m-dependent random variables, ’n and an de8nedas in Theorem 1. If (22) holds, then (28) holds and (29) holds provided an ↑ ∞.

Proof. Noticing that if {Xn; n¿ 0} is a sequence of m-dependent random variables, then E[Xn |F−m] = E(Xn), this corollary follows from Corollary 1.

W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131 127

Corollary 4. Let {Xn;Fn; n¿0} be a martingale di:erence sequence, ’n and an be as in Theorem 1,and let A be de8ned by (2). Then,

∞∑n=1

Xn

anconverges a:e: on A: (30)

If moreover an ↑ ∞, then

limn→∞

1an

n∑i=1

Xi = 0 a:e: on A: (31)

Proof. Noticing that E[Xn |Fn−1] = 0 a.e., this corollary follows from Theorem 1.

Corollary 5 (see Chow and Teicher, 1988, p. 249): Let {Xn;Fn; n¿ 0} be an Lp martingaledi:erence sequence and 0¡an ↑ ∞. Then

limn→∞

1an

n∑i=1

Xi = 0 a:e: on A;

where A= {! :∑∞

n=1 a−pn E[|Xn|p |Fn−1]¡∞} and p∈ [1; 2].

Proof. Letting ’n(x) = |x|p in Corollary 4, this corollary follows.

Corollary 6. Let {Xn; n¿ 0} be a Markov process, fn(x) be measurable functions, ’n(x) be as inTheorem 1 and 0¡an ↑ ∞. If

n∑i=1

E[’n(fn(Xn))]’n(an)

¡+ ∞; (32)

then for any m¿ 1∞∑n=1

1an

{fn(Xn) − E[fn(Xn) |Xn−m; ]} converges a:e: (33)

and

limn→∞

1an

n∑k=1

{fk(Xk) − E[fk(Xk) |Xk−m]} = 0 a:e:; (34)

where X−n; n¿ 1 are constant.

Proof. Let Fn=�(Xn; Xn−1; : : :). Then {fn(Xn);Fn} is a stochastic sequence. Using arguments similarto those used to derive Corollary 1, by (32) we have for any m¿ 1,

∞∑n=1

1an

{fn(Xn) − E[fn(Xn) |Fn−m]} converges a:s: (35)

128 W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131

and

limn→∞

1an

n∑k=1

{fk(Xk) − E[fk(Xn) |Fk−m]} = 0 a:s: (36)

Since {Xn; n¿ 0} is a Markov process, then E[fn(Xn) |Fn−m]=E[fn(Xn) |Xn−m]. Eqs. (33) and (34)follows from (35) and (36) respectively.

If {Xn; n¿ 0} is a nonhomogeneous Markov chain taking values in S={1; 2; : : :}, then Corollary 6also holds. By Corollary 6, we can get Liu and Liu’s result (see Liu and Liu, 1994).

Theorem 2. Let condition (1) of Theorem 1 be replaced by the following condition: as |x| increases,

’n(x) ↑; ’n(x)|x| ↑ (37)

and let A be de8ned by (2). Then (30) holds. If in particular (22) holds, then∞∑n=1

Xn

anconverges a:e: (38)

Proof. Let X ∗n = XnI(|Xn|6 an). Using similar argument used to derive (10), we have

∞∑n=1

1an

(Xn − X ∗n ) converges a:e: on A: (39)

Let Ak and �k be de6ned similar to (5) and (6). By (37)

|X ∗n |=an6’n(X ∗

n )=’n(an)6’n(Xn)=’n(an): (40)

Imitating the proof of (8), we have∞∑n=1

∫Ak

’n(Xn)’n(an)

dP6 k: (41)

By (40) and (41)∫Ak

( ∞∑n=1

|X ∗n |an

)dP =

∞∑n=1

∫Ak

|X ∗n |an

dP6 k:

Hence∑∞

n=1 (X∗n =an) converges absolutely a.e. on Ak , and so it converges a.e. on Ak . Since A=

⋃k Ak ,

we have∞∑n=1

X ∗n

anconverges a:e: on A: (42)

Eq. (30) follows from (39) and (42). Eq. (22) implies P(A) = 1, thus (38) holds.

W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131 129

Corollary (see LoLeve, 1978): Let {Xn; n¿ 0} be a sequence of arbitrary random variables. If thereexist rn ∈ (0; 1] (n = 1; 2; : : :) such that

∑∞n=1 E|Xn|rn ¡ + ∞, then

∑∞n=1 Xn converges absolutely

a.e.

Proof. Considering the sequence of random variables {|Xn|; n¿ 0}, and letting ’n(x) = |x|rn andan = 1 in Theorem 2, this corollary follows.

3. Some strong laws of large numbers

De#nition 1. Let {Xn; n¿ 0} be a sequence of random variables, and let Fmn = �(Xn; Xn+1; : : : ; Xm).

We say that the sequence {Xn; n¿ 0} is *-mixing if there exist a positive integer N and a nonde-creasing function ’(n) de6ned on integers n¿N with limn→∞ ’(n)=0, such that for n¿N; A∈Fm

0and B∈F∞

m+n the relation

|P(AB) − P(A)P(B)| = ((n)P(A)P(B)

holds for any integer m¿ 1.It has been proved that the *-mixing condition is equivalent to the condition

|P(B |Fm0 ) − P(B)|6((n)P(B) a:e:

for B∈F∞m+n and m¿ 1, and implies, for m¿ 1,

|E[Xn+m |Fm0 ] − E(Xn+m)|6((n)E|Xn+m| (43)

with probability 1.

Theorem 3. Let {Xn; n¿ 0} be a *-mixing sequence of random variables. Let Fn = �(X0; : : : ; Xn)and F−n= {�;)}; n¿ 1; ’n de8ned as in Theorem 1. Suppose that E|Xn|6 c; n¿ 1, where c¿ 0is a constant. If

∞∑n=1

E[’n(Xn)]’n(n)

¡+ ∞ (44)

then

limn→∞

1n

n∑k=1

{Xk − E(Xk)} = 0 a:e: (45)

Proof. By (44) and the Corollary 1 of Theorem 1, we have for m¿ 1

limn→∞

1n

n∑k=1

{Xk − E[Xk |Fk−m]} = 0 a:e: (46)

130 W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131

Since {Xn; n¿ 0} is *-mixing, by (43) and E|Xn|6 c; n¿ 1∣∣∣∣∣1nn∑

k=1

{Xk − E(Xk)}∣∣∣∣∣6

∣∣∣∣∣1nn∑

k=1

{Xk − E[Xk |Fk−m]}∣∣∣∣∣+ 1

n

n∑k=1

|E[Xk |Fk−m] − E(Xk)|

6

∣∣∣∣∣1nn∑

k=1

{Xk − E[Xk |Fk−m]}∣∣∣∣∣+ ((m)

1n

n∑k=1

E|Xk |

6

∣∣∣∣∣1nn∑

k=1

{Xk − E[Xk |Fk−m]}∣∣∣∣∣+ ((m)c a:e: (47)

By (46), (47) and ((m) → 0(m → ∞), (45) follows.

Corollary (see Blum et al., 1963): Let {Xn; n¿ 0} be *-mixing sequence such that E(Xn) = 0.Suppose that E|Xn|6 c; n¿ 1, and

∑∞n=1 E(X 2

n )=n2 ¡+ ∞. Let Sn =

∑nk=1 Xk . Then Sn=n

a:e:→ 0.

Proof. Letting ’n(x) = x2 in Theorem 3, this Corollary follows.

De#nition 2. Let {Sn =∑n

k=0 Xk; n¿ 0} be a sequence of random variables, and Fn are adapted toXn. We say that the sequence {Sn;Fn; n¿ 0} is a limit martingale (see, Mucci, 1973) if

limn→∞ sup

m¿n|E[Sm |Fn] − Sn| = 0 a:e:

We say that the sequence {Sn;Fn; n¿ 0} is an adjacent limit martingale if limn→∞ E[Xn+1 |Fn] = 0a.e.

It is easy to see that a limit martingale must be adjacent limit martingale.

Theorem 4. Let {Sn =∑n

k=1 Xk;Fn; n¿ 0} be an adjacent limit martingale, ’n be de8ned as inTheorem 1. Let

A=

{! :

∞∑n=1

E[’n(Xn)]’n(n)

¡+ ∞}: (48)

Then Sn=na:e:→ 0 on A.

Proof. By (48) and Theorem 1, we have

limn→∞

1n

n∑k=1

{Xk − E[Xk |Fk−1]} = 0 a:e: !∈A:

Since

limn→∞

1n

n∑k=1

E[Xk |Fk−1]} = 0 a:e:;

this theorem follows.

W. Liu, W. Yang / Statistics & Probability Letters 64 (2003) 121–131 131

The authors would like to thank the referees for their valuable suggestions.

References

Blum, J.R., Hanson, D.L., Koopmans, L., 1963. On the strong law of large numbers for a class of stochastic processes.Z. Wahrsch. Verw. Gebiete 2, 1–11.

Chow, Y.S., Teicher, H., 1988. Probability Theory, 2nd Edition. Springer, New York.Chung, K.L., 1974. A Course in Probability Theory, 2nd Edition. Academic Press, New York.Liu, W., 1990. Relative entropy densities and a class of limit theorems of the sequence of m-valued random variables.

Ann. Probab. 18, 829–839.Liu, G.X., Liu, W., 1994. On the strong law of large numbers for functional of countable nonhomogeneous Markov chains.

Stochastic Processes Appl. 50, 375–391.LoLeve, M., 1978. Probability Theory, 4th Edition. Springer, Berlin, New York.Mucci, A.G., 1973. Limits for martingale-like sequence. Paci6c J. Math. 48, 197–202.Petrov, V.V., 1975. Sums of Independent Random Variables. Springer, New York.