ON LARGE DEVIATION PROBABILITIES FOR PROPERLY NORMALIZED WEIGHTED SUMS AND RELATED LAW OF ITERATED L

ON LARGE DEVIATION PROBABILITIES FOR PROPERLY NORMALIZED WEIGHTED SUMS AND RELATED LAW OF ITERATED LOGARITHM

* GOOTY DIVANJI 1 | K. VIDYALAXMI 2 1 Department of Studies in Statistics, Manasagangotri, University of Mysore, Mysuru - 570006, Karnataka, India. (*Corresponding Author) 2 Department of Community Medicine, JSS Medical College, Mysuru - 570015, Karnataka- India.

ABSTRACT

Let X n , n  1 be a sequence of independent and identically distributed random variables with distribution function F. When F belongs to the domain of attraction of a stable law with index α, 0 <α < 2 and α≠1, an asymptotic behaviour of the large deviation probabilities with respect to properly normalized weighted sums have been studied and in support of this we obtained Chover’s form of law of iterated logarithm.

Keywords: Large deviation probability, weighted sum, law of iterated logarithm

where x n , n  1 is a monotone sequence of positive

1.INTRODUCTION Let

Xn , n  1 be

a sequence of independent and

identically distributed (i.i.d) random variables (r.v.s) with distribution function (d.f) F, which belongs to the domain of attraction of a stable law with index α, 0<α<2 and α≠1. We denote this as FϵDA(α),0<α<2 and α≠1. n

n

k=1

k=1

Set Sn = X k, n  1 and Tn = f  kn X k, where f is a non-decreasing and continuous on [0,1] and for any xϵ[0,1], f(x)=1 gives us partial sums. Let Bn = inf x > 0 : 1- F(x) + F(-x)  n1  . Since F ϵ 1 α

DA(α), 0<α<2 and α≠1, then we can have Bn = n l(n) , where l is a function slowly varying (s.v) at ∞. When F ϵ DA(α),0<α<2 and α≠1, Beuerman [1975] proved that

Lim P

n

  = G(x) ,where G is a limiting stable law T Bn

with index α, 0<α<2 and α≠1.Probability of large values plays an important role in studying the non-trivial limit behavior of stable like r.v.s. As far as properly normalized partial sums of stable like r.v.s, we can use the asymptotic results of Heyde [1968] for obtaining law of iterated logarithms (LIL) or rate of convergence problems [See Vasudeva [1978] and Gooty Divanji [2004]]. It is well known that the probabilities of the type P |Sn | > x n  or either of the one sided

numbers with x n   as n→∞, such that

Sn xn

P  0

as n→∞. In fact, under different conditions on the sequence of r.v.s, Heyde [[1967a],[1967b] and [1968]] studied large deviation problems for partial sums. In brief, when the underlying r.v.s are in the domain of attraction of a stable law, with index α, α≠1, Heyde [1968] obtained the precise asymptotic behaviour of large deviation probabilities.

For unit variance, Allan Gut [1986] studied the classical LIL for geometrically fast increasing subsequences of (Sn ). In fact, he established that

limsup k

Sn k n k log log n k

 2 a.s., if limsup nnk+1 < k  k = , n k+1 * ε a.s., if liminf n k >1 k 



-ε 2  2 log n   k  <   . Torrang k=1 



where ε* = inf ε > 0 :



[1987] extended to random subsequences.

n k =2 2 , then ε*=0 and we have

that, when

lim sup k 

Observe

k ..

Sn k n k loglog n k

norming sequence

= 0 a.s. i.e, for such cases the n k log log n k will not be precise

components are called large deviation probabilities,

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enough to give almost sure bound for (Sn k ) . In general whenever

n k+1  , as k  , Schwabe and Gut nk

[1996] have pointed out that

n k loglogn k is not a

proper normalizing sequence and it has to be replaced by

n k log k . Note that nnk+1k  , as k  , comes

under the class of at least geometrically fast increasing subsequenes.

When nk=n, Chover [1966] observed that in the case of stable r.v.s, LIL involving lim sup cannot be obtained under linear normalization and that it is possible under power normalization only. In fact, when Xn's are i.i.d. symmetric stable r.v.s, Chover [1966] established the LIL for (Sn), by normalizing in the power i.e., 1 loglogn

lim sup n  

Sn 1 nα

1

= e α a.s .

Peng and Qi [2003] obtained

Chover’s type LIL for weighted sums of i.i.d r.v.s which are in the domain of attraction of a stable law with index α, 0<α<2, where the weights belong to Bounded Variation on [0,1], we denote the same as BV[0,1].

Many authors studied the non-trivial limit behavior for weighted sums. (See Vasudeva [1978] and Peng and Qi [2003]).

However, the observations made by Heyde [1967b] on the large deviation probabilities implicitly motivated us to study the large deviation probabilities for weighted sums and obtain a non-trivial limit behavior of properly normalized weighted sums for subsequences.

In the next section we present some lemmas and main results are presented in section 3. In the last section, we discuss the existence of Chover's LIL for weighted sums for subsequences. In the process, i.o, a.s and s.v mean 'infinitely often', 'almost surely' and 'slowly varying' respectively. C, ε, k and n with or without a super script or subscript denote positive constants with k and n confined to be integers.

2. Lemmas

-δ L(x n y n ) lim yn L(x ) = 0 . n  n

Lemma 2 [Vasudeva and Divanji [1991]] Let F ϵ DA(α),0<α<2. Let (xn) be a monotone sequence of real numbers tending to ∞, as n→∞. Then

Sn P   0, as n  , where x n Bn 1 α

Bn = n l(n) and l is s.v. at ∞.

Lemma 3 Let F ϵ DA(α),0 <α< 2 and α≠1. Let (xn) be a monotone sequence of real numbers tending to ∞, as n→∞. Then 1

Tn P   0, as n  , where Bn = n α l(n) and l x n Bn n

is s.v. at ∞ and Tn =

Let L be any s.v. function and let (xn) and (yn) be sequences of real constants tending to ∞ as n→∞. Then

L(x n yn ) for any δ >0, lim y =  and n  L(x n ) δ n

k

k=1

and continuous function on [0,1].

Proof Since f is a non-decreasing and continuous function on [0,1] and by Abel’s partial summation formula, we have

 f  kn  X k =  f  kn  -f  k-1n   Sk + f(0)S0 n

n

k=1

k=1

 max Sk  f(1) - f(0)  1 k  n

 Sn  f(1) - f(0) 

(1)

We have n k Tn = f   X k  Sn  f(1)-f(0)  = Sn . Dividing on k=1  n  Tn S both sides by xnBn, we have  n . By x n Bn x n Bn

Lemma 2, we have

Lemma 1 [Drasin and Seneta [1986]]

k

 f  n  X , f is a non-decreasing

Sn P   0 and this leads to x n Bn

Tn P   0, as n  . x n Bn

3. Main results Theorem 1 Let X n , n  1 be a sequence of i.i.d r.v.s with a d.f F and assume that FϵDA(α),0<α<2 and α≠1. Let (xn) be a

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monotone sequence of positive numbers with xn →∞, as n→∞.

P Tn  x n Bn  n   nP X  x n Bn 

Then

Lim

= 1,

where

Note that P  Tn  x n Bn   P  T1,n > x n Bn  + P  T2,n  0  .

n

Tn = f  kn  Xk, and

f

is

non-decreasing

and

k=1

continuous function on[0,1] with f(1)=1 and f(0)=0, 1 α

P  Tn  x n Bn 

Bn = n l(n) and l is s.v. at ∞.

nP  X  x n Bn 

Proof To

prove

Lim inf n

the

assertion,

P Tn  x n Bn  nP X  x n Bn 

This implies

first

we

show



P  T1,n  x n Bn 

nP  X  x n Bn 

 1 and later we establish that

P T x B

n

P  Tn  x n Bn   P(Sn  f(1)-f(0)   x n Bn )

(3)

P  T2,n  0 

that

Lim sup nP  Xn  xnn Bnn   1 Observe that by (1), we have



nP  X  x n Bn 

Observe that

 n  P  T2,n  0   P   R k  0   k 1 

 P(Sn   f(1)-f(0)  x n Bn )

n

  P(R k  0)

-1

k 1

which implies P Tn  x n Bn  nP X x n Bn 





P Sn  f(1)-f(0)  x n Bn -1

nP X  x n Bn 

 . It

n  k    P  f   X k  x n Bn . k 1   n  

is well known that

when F ϵ DA(α),0<α<2 and α≠1, [See Mijnheer [1975] of Theorem 2.2 on page 16], we have

Since f is non-decreasing and continuous on [0,1] and for all k, 1 ≤ k ≤ n, we may get that

Lim 

1 k 1 f    f    f(1) or f    f(1) . n n n

1- F(x) + F(-x) 

x 

x-

α

L(x)

=1

(2)

(4)

Hence using (2), we get, P Tn  x n Bn 

nP X  x n Bn 





P Sn  f(1)-f(0)  x n Bn

 

-1

nP  X  x n Bn 

n  f(1)-f(0)  x n Bn -1

n  x n Bn 









Using the fact (4), one can find some k such that for all k ≥ k1, we have



L  f(1)-f(0)  x n Bn -1

L  x n Bn 



k=1



  f(1)-f(0)  , -α

n

P  Tn  x n Bn  nP  X  x n Bn 





 nP X  f -1 (1)x n Bn ,

for large n (by Lemma 1). Since f is non-decreasing and continuous on [0,1] and for large n, we can get that

Lim inf



n

P  T2,n  0    P X k  f -1 (1)x n Bn

 1.

since Xk’s are i.i.dr.v.s. Using (2), we have

P  T2,n  0 

nP  X  x n Bn 

In order to complete the proof, we use truncation method. Define

if f  kn  X k  x n Bn X , Yk =  k . otherwise 0, Let R k = f  kn  X k - f  kn  Yk , n

n

k=1

k=1





nP X  f -1 (1)x n Bn nP  X  x n Bn 

 f (1) α



L f -1 (1)x n Bn L  x n Bn 





Again using Karamata's representation of s.v. function at ∞, one gets that

T1,n = f  kn  Yk and T2,n = R k .

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

By Theorem 1 on page 544 of Feller [1986, Vol. II] and (2), one gets that

L(x n Bn ) x n Bn f (1)x n Bn ε(y) = a(x n Bn ) exp   y dy   0  0 -1 f (1)x n Bn ε(y)  a  f -1 (1)x n Bn   a(x n Bn ) exp   y dy   x n Bn 



a f -1 (1)x n Bn

-1



ε(y) y

 dy  

k=1

nx 2n Bn2 P(X  x n Bn ) x αn Bαn  f 2



 C2 , ε(y)  δ0 , for y  x n Bn . This

a(x n Bn )

n

n

Since ε(y) → 0 and a(y) → C < ∞ as y → ∞, there exists C2 > 0 and δ0< α, such that

a  f -1 (1)x n Bn 

 f 2  kn  EYk2

k=1

nx 2n Bn2 L(x n Bn )

1 n   f n k=1

  x B    L(x B ) L f -1

k n



  C exp 2

L(x n Bn )



δ log  f

-1

0

(1)



C2 . f (1)

Using the fact that f is non-decreasing and continuous on [0,1] and for some constant C3, one gets that P T  0 

Lim nP X2,n x n Bn   Lim  f(1)  n 

α-δ0

n 

 C3

(5)

Now consider the first term in the right of (3). By Tchebychev’s inequality, we get,

P  Tn  x n Bn 

nP  X  x n Bn 



E T

2 1,n



nx B P(X  x n Bn ) 2 n

=

2 n



 exp  f



  0

a  x n Bn 

a f

1

-1

 exp  f

(1)x n Bn

 (1)x n Bn ε(y) x B ε(y)  dy- n n dy  y y 0 

-1





 (1)x n Bn ε(y)  dy  y 

  x n Bn

a  x n Bn 



Since ε(y) → 0 and a(y) → C < ∞ as y → ∞, there exists C4> 0 and δ0< α, such that



a f 1 (1)x n Bn a  x n Bn 





 C4 ,ε(y)  - δ0 , for y  x n Bn . This gives

L f 1 (1)x n Bn L  x n Bn 





C4 exp -δ0 log

. Since

n k 2 E(T1,n ) =  f 2  EYk2 + n k=1 n n k m f   f   EYk EYm  k=1 m=1  n   n  km

We have

L  x n Bn 



a f -1(1)x n Bn



δ0

n n

Using Karamata’s representation of s.v. function at ∞, one gets that



L f -1 (1)x n Bn

k n

n n

L f -1 1 x n Bn

yields

2-α -1 k  kn  f α-2  kn  x 2-α n Bn L  f  n  x n Bn 

   C  f 1 f(1)

δ0

1

(1)



(7) From (7), one can find a constant C5 (> 0) such that

k 2   EYk C n k=1  5 2 2 nx n Bn P  X  x n Bn  n n

f

2

n

f  

0

k=1

k   n

By the assumption (4), we can find a constant C6 (> C5)

k 2   EYk n k=1  C6 such that 2 2 nx n Bn P  X  x n Bn  n

P  T1,n  x n Bn 



nP  X  x n Bn 

f

 

2 E T1,n

nx n2 Bn2 P(X  x n Bn )

k 2   EYk n    k=1 nx 2n Bn2 P(X  x n Bn )

2

n

f

2

(8) Observe that

k m f   f   EYk EYm  n k=1 m=1  n  km  nx 2n Bn2 P(X  x n Bn ) n

n

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k m

n

 k f -1   x n Bn  n n  δ δ k x -α-δ0 dx x n0 Bn0   C7 1+δ0   f n k=1 0    D 2-α nx 2-α n Bn

 f  n  f  n  EY EY k

 

m

k=1 m=1

km  n k    f   |EYk |  k=1  n  

2

k f -1   x n Bn n

Now for 0 <  < 1, |EYk |  E|Yk | =





Hence



|x|dP(X<x)

there

exists

P(X  x)dx Lemma

that

1,

and using

we

get,

DA(α), 0 < α <2 and α ≠ 1, we know that, for some C9>

nL(Bn )  C9 and choose δ< α such that there exists Bαn L(x n Bn ) some constant C10 (> C9) such that  C10 . nxn Bαn 0,

2

 n k   f   |EYk | n  B   k=1 2 2 nx n Bn P  X  x n Bn 

2

  k  Therefore D  C10   f α+δ0    . Since f(x) is non n  k=1  n

2

Note that A ≤ B ≤ D. Again using (2), we get, 2

decreasing and continuous on [0,1], one can find C11 n

f

such that

α+δ0

k=1

k    nC11 and hence for some C12 n

(> C11), we have

D  C12  B  C12  A  C12 .

(9)

On the other hand, if 1 < α < 2 and EX1=0 then

|EYk |= 2

k   f -1   x n Bn n n   -α L ( x ) k   f  n   x L xn Bn  dx  0  k 1    L xB .   n n 2- 2- nx n Bn







k f -1   x n Bn n



k f -1   x n Bn n

xdF(x) .

L(x) x B   C7 1+δ0   n n  . Substituting these facts L(x n Bn )  x  we get,

Majoring

|EYk|

by

P  X  x dx and following similar steps of

the case 0 < α < 1, we can able to show that n

n

k m

 f  n  f  n  EY EY k

Following similar steps of (5), we can find some constant C7and δ0>0 such that 0

such

L(x n Bn ) L(Bn )  nL(Bn ) 1  α α xn  . Since F ϵ nx αn Bαn nx n Bn Bαn n 2 x α-δ n

k m f   f   EYk EYm  n k=1 m=1  n  km Let A  , nx 2n Bn2 P  X  x n Bn  n

 n  k  f -1  kn  x n Bn    f   0   P(X  x)dx  n k 1  and D   2 2 nx n Bn P  X  x n Bn 

C7)

L(x n Bn ) L(Bn ) L(x n Bn ) = α α nx n Bn nx αn Bαn L(Bn )

Consider

k   f -1   x n Bn n n   -α k   f  n   x L(x)dx  0  k 1    D nx 2-n  B2-n  L  xn Bn 

C8(>

 n k  C8   f α+δ0    L(x n Bn ) n  .

D   k=1 nx αn Bαn

0

n

0 0 α+δ 0 -1 x1-α-δ B1-α-δ f  nk  . n n

1 1-α-δ0

2

k |x| f   x n Bn n



x -α-δ0 dx =

0

-1 

k f -1   x n Bn n



Since

2

       L x B  n n

m

k=1 m=1

km nx 2n Bn2 P  X  x n Bn 

INTERNATIONAL EDUCATION & RESEARCH JOURNAL [IERJ]

 0 as n  

(10) 57

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From

(8),

(9)

P  T1,n  x n Bn  nP  X  x n Bn 

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 8 | August 2016

and

(10),

we

claim

that,

 0, as n   .

Lim sup n 

(5)

(13)

To prove (12),

(11) Substituting

1 ε* -ε   α P  Tn k  n k  logn k  α i.o   1  

and

P  Tn  x n Bn 

nP  X  x n Bn 

(111)

in

(3),

we

get

 1, and the proof of the

theorem is completed.

1 ε* +ε   α let M k  Tn k  n k  log n k  α  and  

ynk  n

1 α k

 log n k 

ε* +ε α

.

By the Theorem 1, one can find a C1 and a k1 such that, for all k  k1 , CHOVER’S FORM OF LIL FOR

P(M k )  C1n k P(X  y n k ).

SUBSEQUENCE OF PROPERLY

known that

NORMALIZEDWEIGHTED SUMS

F  DA  α  , 0 < α < 1, the equation (2)

Theorem 2 Let X n , n  1 be a sequence of i.i.d positive r.v.s with a d.f F and assume that F ϵ DA(α),0<α<1. Let

 k  Tn k =  f   X n k , where f is a positive, nonk=1  n k  n

decreasing and continuous function on [0,1] . Let {nk} be an integer subsequence such that liminf k 

T Lim sup  n1k n   n kα Then

n k+1 nk

1.

1

 log log nk ε*  e α a.s,      -ε * where ε = inf ε  0 :   log n k     . k=k 0  

n In particular, if Lim k+1  , as k  , but not limit k  n k is ∞ and ε*=0, then

T Lim sup  n1k n   n kα

n 

1-F(x)  1 and using this fact, one can x -α L(x)

find a k2 ( ≥ k1) such that for all k ≥ k2,

P(M k )  C1n k y-αn k L(y n k )  1 L  n kα  C1n k L(y n k )  .  * (ε +ε)  α1  n k  logn k  L  nk    Using Lemma 1, with δ=

ε , 2

we choose ε

sufficiently small and by the definition of ε*, one can find k3(≥ k2) such that for all k(≥ k3),

P(Mk )  C2 (logn k )



- ε* + 2ε



for

some

C2

>

0.



 P(M )< k

k  k3

1

and (13) follows from

Borel-Cantelli lemma.

Using the relation Tnk =Tnk -Tnk-1 +Tnk-1 ,k  1and define,



(k-1) for large k, m k = min j:n j  β

Proof To prove the assertion, it suffices to show for any ε ϵ (0,1) that

and

becomes Lim

Consequently,

 log k 1  = e α a.s. 

1 ε* +ε   α P  Tn k  n k  logn k  α i.o   0  

It is well

(12)

δ



(14)

where β>1 and δ>0. In order to establish (13), it is enough, if we show that for εϵ(0,ε*), 1 ε -ε   P  Tn m -Tn  2n mα k (logn mk ) α i.o  = 1  k mk-1   

INTERNATIONAL EDUCATION & RESEARCH JOURNAL [IERJ]

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(15)

58

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 8 | August 2016 1 (ε -ε)   α P  Tn m  n mk (log n mk ) α   k-1    . n mk-1 1  C6  * 3ε  n mk ε -  (log n mk ) 2 

and

*

1 ε +ε   P  Tn  n mα k (logn mk ) α i.o  = 0  mk-1    *

1 α

Define z n =n (logn)

(ε* -ε) α



(16)

and



D k = Tn m -Tn m  z n m , k  1. Note that k k-1 k

k

k-1

=T

n mk -n m

,  k  1. Hence by Theorem 1,



P(D k )  C3 (n mk -n mk-1 )P X  2z n m  nm  C3 n mk 1- k-1  nm k 

Since liminf k 

n k+1 nk

n mk

k



exists λ>1 such that nk+1≥λnk. Therefore,

  P X  2z n mk 



< λ < 1, for



P(D k )  C3n mk P X  2z n m

k

all

δ

n mk+1  β k  n mk  λn mk-1  δ

δ

δ

λn mk-1  β k  n mk-1  λ -1β k =λ1β k ,



1 where λ1 = . λ

 1 implies that there exists λ<1 such

n mk-1

that

implies

k-1

one can find a constant C3>0 and k4 such that for all k

(≥ k4),

δ

n k+1  1, there nk

δ

n mk+1  β k  n mk and since lim inf k 

d

Tnm -Tnm

n mk  β(k-1)

From (14), we infer that

k≥k4,

Hence

n m k-1 n mk



λ1β k

δ

β(k-1)



δ

λ1 βk

δ1

. We choose ε sufficiently small

and by the definition of ε*, we get

.



Now following the similar steps to those used to get an upper bound (13), one can find C4> 0 and k5 such that for all k (≥ k5),

 k=k 7



n mk-1 n mk

1  * 3ε  ε -  2 

 log n mk 

 λ1 

k=k 7 β

1 k δ1

 * 3ε  ε -  2

.

(logn mk )

Therefore 1 (ε -ε)   P  Tn m  n mα k (log n mk ) α i.o   0, which implies  k-1    *

P  Dk   C4 (log n mk ) 

  log n k 

ε  - ε* -  2 

 * ε ε -  2 

.

Note

that

  . Since Dk’s are mutually

k=k 4



independent and

 P(D )   and by Borel-Cantelli k

k=k 4

lemma, we establish (15). a

k6

Again using Theorem 1, one can find a C5 and such that, for all k ≥ k6,

1 (ε -ε)   P  Tn m  n mα k (log n mk ) α   k-1    1 (ε* -ε)   α  C5 n mk-1 P  X1  n mk (log n mk ) α      *

Again following the steps similar to those used to get a lower bound of P(Mk), one can find a constant C6>0 and k7 such that, for all k(≥ k7),

(13) follows from the proofs of (15) and (16) which completes the proof.

To prove (12), it suffices to show for any ε2, 0 < ε2< 1, that, 1 (1+ε 2 )   P  Tn k  n kα k α i.o   0  

(17)

and 1 (1-ε 2 )   P  Tn k  n kα k α i.o   1  

(18)

Observe that as the case

n k+1 = , as k→∞ comes under the class of at least k  n k lim

geometrically increasingsubsequences, the proofs of

INTERNATIONAL EDUCATION & RESEARCH JOURNAL [IERJ]

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Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 8 | August 2016

(17) and (18) follows on the similar lines of proofs of (15) and (16) and hence the details are omitted.

 

nv 1 θ  j-1  <1. Consequently, nv j satisfies the θM n v j M

Theorem 3 Let

Xn , n  1

be a sequence of i.i.d positive r.v.s

condition Lim sup j

with a d.f F and assume that F ϵ DA(α), 0 < α < 1. Let

 k Tn k =  f  k=1  n k n

  X n k , where f is a positive, non

decreasing and continuous function on [0,1]. Let {nk} be an integer subsequence such that Lim sup k 

n k+1 <. nk

Then

  log n 

relation

vj

j=1

n v j-1 nv j



-ε1

< 1 of Theorem and also the

<  holds for all ε1>1 (i.e., ε*

=1). Now (20) follows from the Theorem. Hence the proof of the theorem is completed.

References 1. Allan Gut (1986): Law of iterated logarithm for subsequences, Probab.

 Tn Lim sup  1k  n  nα  k

1

Math-Statist. 7(1), 27-58.

 log logn k 1  = e α a.s.   

2. Beuerman, D.R (1975): Limit distributions for sums of weighted random variables, Canad.Math.Bull.18(2) 3. Chover, J (1966): A law of iterated logarithm for stable summands, Proc.

Proof

Amer. Math. Soc.17,441-443.

Proceeding as in Theorem, it is enough if we show that for any ε1ϵ(0,1),

4. Drasin, D and Seneta, E (1986): A generalization of slowly varying

1 1+ε1   α P  Tn k  n k  logn k  α i.o  = 0  

5. Gooty Divanji (2004): Law of iterated logarithm for subsequences of

(19)

functions. Proc.Amer.Math.Vol 96,470-472.

partial sums which are in thedomain of partial attraction of semi stable law, Probability and Mathematical Statistics, Vol.24, Fasc. 2,41, pp. 433-442.

and

6. Heyde.C.C (1967a): A contribution to the theory of large deviations for

1  P  Tn k  n kα  logn k 

1-ε1

α

 i.o  = 1 

sums of independentrandom variables, Zeitschrift. fur Wahr. Und ver. Geb,

(20)

band 7, 303-308.

7. Heyde.C.C (1967b): On large deviation problems for sums of random

One can note that (19) is a consequence of the theorem of Vasudeva[1978], that is

1575-1578.

1

 T  log log nk Lim sup  n1k  k   n k 

8. Heyde.C.C (1968): On large deviation probabilities in the case of

 Lim sup n 

a

sequence

of

  Tn Bn

where Bn is 1 log log n

constants

1 α

= e a.s. with

1  Bn = inf x > 0:1-F(x)+F(-x)   . n  Lim sup k 

variables which are notattracted to the normal law, Ann. Math. Statist. 38(5),

Bn>

attraction to a non normal stable law,Sankhya, ser. A, 30, 253-258.

9. Ingrid Torrang (1987): Law of iterated logarithm - Cluster points of

0

and

deterministic and random subsequences, Prob.Math. Statist. 8, 133-141.

10. Liang Peng and Yongcheng Qi (2003): Chover-type laws of the iterated

Since

n k+1 <  we see that the sequences are at most nk

geometrically increasing, which implies that there exists ϴ>1 such that n k+1  θn k . Now define where M is

θ <1. Proceeding as in Allan Gut chosen such that M j j [1986], one can show that M < n v j < θM and

logarithm for weightedsums, Statistics and Probability letters, 65, 401-410.

11. Rainer Schwabe and Allan Gut (1996): On the law of the iterated logarithm for rapidly increasingsubsequences. Math.Nachr. 178, 309-332.

12. Vasudeva, R (1984): Chover's law of iterated logarithm and weak convergence, ActaMath.Hungar. 44(3-4), 215-221.

13. Vasudeva, R and Divanji, G (1991): Law of iterated logarithm for random subsequences, Statistics and Probability letters 12, 189-194.

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