ALMOST SURE LIMIT POINTS FOR DELAYED RANDOM SUMS

Research Paper

Statistics

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

ALMOST SURE LIMIT POINTS FOR DELAYED RANDOM SUMS

K. N. Raviprakash 1 | Gooty Divanji 1 Department of Stidies in Statistics, Manasagangotri, University of Mysore, Mysuru-570 006, Karnataka, India.

ABSTRACT

X n , n  1 be a sequence of

Let

i.i.d. positive asymmetric stable random variables with a common distribution function F with index

α , 0  α  1.

The present work intends to obtain almost sure limit points for a sequence of properly normalized delayed random sums.

KEYWORDS: Law of iterated logarithm, Delayed sums, Delayed random sums, Asymmetric stable law, Almost sure limit points.

1. INTRODUCTION Let X n , n  1 be a sequence of independent and identically distributed (i.i.d.) positive asymmetric stable random variables (r.v.s) with a common distribution function (d.f.) F with index

α , 0  α  1 .Set

n

Sn   X k ,

n  1 and

limsupinf Yn  α(β) is

to

be

read

as

limsupYn  α and

liminfYn  β .

We will frequently use the the following well known results.

k 1

Ta  n

n a n

X

i  n 1

i

 Sn a n  Sn

, where

a n , n  1 is a non-decreasing sequence

of the positive integers of n such that, where

bn

0  a n  n , for all n and

is non-increasing. The sequence

T

an

an ~ bn , n

 is called a (forward)

,n 1

2. SOME KNOWN RESULTS Lemma 1 (Extended Borel-Cantelli Lemma) Let

E n  be a sequence of events in a common probability space. 

If

(i)

 P E n   

N n , n  1

Nn

 e n  1  0 almost surely as

n

random sums as, M N  n

Xn 's

n  Nn

 X j  Sn Nn  Sn

.

are i.i.d. symmetric stable r.v.s, with index

α ,0  α  2

Chover (1966) studied the law of iterated logarithm (LIL) for

Sn  ,

by

normalizing in the power. For further developments in Chover’s form of LIL see GootyDivanji (2004).

When variance is finite, Lai (1973) had studied the behavior of classical LIL for

T  , at different values of an

k 1

s 1

liminf

k

 Es 

    PE k   k 1  n

2

C,

E n i.o.  C 1.

then P

Where C is some positive constant.

j n 1

properly normalized sums

n

n 

n   , where

0    1 . Now parallel to the delayed sums Ta , we introduce delayed

When

(ii)

be a sequence of positive r.v.s. independent of

X n , n  1such that

n

  PE

delayed sum sequence [See Lai(1973)].

Let

and

n 1

a n 's .

For

 

For proof, see Spitzer (1964, Lemma p3,p.317) Lemma 2

A n  be a sequence of events in a common probability space. If PA n   0 as n   and  PA  A    . Then

Let



PA n i.o.  0 .

n 1

independent, but not identically distributed strictly positive stable r.v.sVasudeva and Divanji (1993) studied the non-trivial limit behavior of delayed sums T

For proof, see Nielsen (1961, Lemma 1*,p.385).

.

Lemma 3

n

c n 1

an

In this In this work, we intend to obtain almost sure limit points for

M  .

Nn

Throughout this Paper, C,  (small), k(integer), with or without a suffix or super suffix stand for positive constants., whereas a.s. and i.o. mean almost sure and infinitely often respectively. For any sequence Yn  of r.v.s

X n , n  1be a sequence of i.i.d. positive asymmetric stable r.v.s with common d.f. F with index α , 0  α  1 . Let N n , n  1be a sequence Let

Copyright© 2016, IERJ. This open-access article is published under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License which permits Share (copy and redistribute the material in any medium or format) and Adapt (remix, transform, and build upon the material) under the Attribution-NonCommercial terms.

International Educational Scientific Research Journal [IESRJ]

29

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

of positive r.v.s independent of

X n , n  1such that

Nn 

e n  1  0 a.s. as

n   , where 0    1 .Let γ n   log n  log logn  . Then   N 

  MN lim inf (sup) 1n n   N  n

    



n

3. ALMOST SURE LIMIT POINTS FOR DELAYED RANDOM SUMS Theorem 1

X n , n  1be a sequence of i.i.d. positive asymmetric stable r.v.s, with common d.f. F with index α , 0  α  1 . Let N n , n  1be a sequence Let

of positive r.v.s, independent of

X n , n  1such that

n

,

Nn

e

n

1  0

0   1

where

.Let

1   n γ n   log  log logn  . Then all the points in 1, e   are a.s. limit Nn    

points

of

the

sequence

  M N  n  1  N   n  

1  n     , n  1     

,

where,

e n 1  ε

implies

u n  N n  v n a.s.,(3)

u n  C1 n δ C 2  log (1  ε) .

We

know

n

MN  n

N n , n  1 X n , n  1,

p  α1  α 1, e  be e , 0  p  1 . Observe that for p=0  

and p=1, the results follow from Lemma 3. Hence, it is enough to show that for

p  (0,1) there exits sufficiently small ε 1  0 ,

X

j n 1

j

C1  log (1  ε)

 Sn  Nn  Sn  S N n

.

and

Since

be a sequence of positive valued r.v.s. independent of by (3) we have

Mu  MN  Mv n

n

n

a.s.

(4)

Hence (1) and (2) hold whenever, p  ε1   1 α     n α P M v  N n  log n  i.o.  0, n  Nn     

where M v

(5)

 S n  vn  S n and

n

Where

(6)

M u  Sn u n  Sn . n

Using (3), one can find some constants

C 3 ( 0)

and

C 4 ( 0)

such

that,

(1)

1 α n

 n  N  log n   Nn 

p  ε1 α

 C3n

(1 δ) (p  ε1 )  δ α

log n 

p  ε1 α

log n 

p  ε1 α ,

and

and p  ε1   1 α     n P M N  N nα  log n  i.o.  1 n  Nn     

n  Nn

,

S u  S N  S v a.s. which implies n n n

Proof

p  ε1   1 α     n α P M N  N n  log n  i.o.  0 n  Nn     

that

vn  C2n δ

,

p  ε1   1 α     n P M u  N nα  log n  i.o.  1, n  Nn     

M N  SnNn  Sn .

Let an arbitrary point in

  (0,1) such that,

Nn

where

In the next section, the almost sure limit points for delayed random sums are obtained.

as

and

 e n  1  0 a.s. as n   implies that there exists

1

 1 (e  ) a.s.

For proof, see Gooty Divanji and K .N. Ravi Prakash (2016 Theorem 1 and Theorem 2).

a.s.

ε0

some δ

1

n

Nn

From the condition

(2)

International Educational Scientific Research Journal [IESRJ]

1 α n

 n  N  log n   Nn 

p  ε1 α

 C4n

(1 δ) (p  ε1 )  δ α

which in turn (5) and (6) implies that,

30

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

(1 δ) (p  ε1 )  δ p  ε1   α  P M v  C 3 n log n  α i.o.  0 n  

A

(7)

(8)

A

(1 δ) (p  ε1 )  δ p  ε1  α log n  1 α  M vn 1  C 3 n  1  

n 1

As

(1δ) (p  ε1 )  δ p  ε1   α   M v  C 3 n  1 log n  1 α  n 1  

Xn 's

 .

x n  C3n

Mv n .

xn

n

  0 p

as

1 α

xv

n  1  ε  , for some ε  0 , Hyede’s xn Theorem (1967), we can find C 5 ( 0) such that,

Also since

1  limsup n 

PA n   C 5 n PX1  x n  .

Using (9), there exits some constant

PA n   C 6 ~ and since

n 1

p  ε1 α

  





which yields,

P An  Ac

n 1

(1 δ) (p  ε1 )  δ α

log n  1

p  ε1 α

  

p  ε1 α

  

(1 δ) (p  ε1 )  δ α

log n  1

p  ε1 α

  

(1 δ) (p  ε1 )  δ p  ε1  α  C3n log n  α M vn  P  1 1    v n α v n α 

C 3 n  1

n

Using the fact that

(1 δ) (p  ε1 )  δ α

log n  1

v n α 1

Mv

p  ε1

(11)

log n  1



 C 3 n  1



C 6 ( C 5 ) such that,

p  ε1

(1 δ) (p  ε1 )  δ α

(1 δ) (p  ε1 )  δ p  ε1  α log n  α  M vn  P C 3 n 

(10)

n (1δ) (p  ε1 )  δ log n  C6

n log n 

 Ac

 C 3 n  1

p  ε1 α . Following proof of Lemma C in

Vasudeva and Divanji (1991), we can show that,

log n  1

(1 δ) (p  ε1 )  δ p  ε1  α log n  α M vn  C 3 n 

(9)

log n 

(1 δ) (p  ε1 )  δ α

n

In view of (9), we can observe that, condition (2) of Heyde (1967) is satisfied by (1 δ) (p  ε1 )  δ α

n

M v  C 3 n  1

are i.i.d. positive asymmetric stable r.v.s., we have,

PX  x n  ~ O x  α

n 1

 C 3 n  1

(1 δ) (p  ε1 )  δ p  ε1  α   M v  C 3 n log n  α , n 

or

Ac



Observe that,

A n 1  and

n 1

Mv

(1 δ) (p  ε1 )  δ p  ε1   α  A n  M v  C 3 n log n  α  n  

Let

 Ac

(1 δ) (p  ε1 )  δ p  ε1  α log n  α , M v n  C 3 n 

and (1 δ) (p  ε1 )  δ p  ε1   α  P M u  C 4 n log n  α i.o.  1. n  

n

n 1 α

v n 

p  ε1 α

   

d

 X1 we have

p  ε  1, PA n   0 as n   .

1

we have

International Educational Scientific Research Journal [IESRJ]

31

Research Paper



P An  Ac

n 1

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016



y n α  y n1   C v n  (1δ) (p  ε )  δ10 1 log n pε1 n

(1 δ) (p  ε1 )  δ p  ε1  α  C3n log n  α  P  X1 1  v n  α 



C 3 n  1

(1 δ) (p  ε1 )  δ α

v n 

 Py n  X 1  y n 1  

log n  1

p  ε1 α

1 α



   

n (1δ) (p  ε1 )  δ log n 

 f(x)dx,



yn

Where

yn 

y n 1 

C3 n

v n 

C 3 n  1

log n 

p  ε1 α

1 α

(1 δ) (p  ε1 )  δ α

v n 

log n  1 1 α

~

p  ε1 α

.

n  1

(1 δ) (p  ε1 )  δ

n (1δ) (p  ε1 )  δ log n  C11 n (1δ) (p  ε1 ) log n 

for some C11>0 and C12>0.



1   1 f(x)  C  1α  1 2α  . x x 



  C   x1

1 α



yn

1   dx x 



1 2α

C  1 1  1 1 1         2  2  .   y n y n 1  2  y n y n 1 

(12)



n 1

y



p  ε1

.

n

2(1δ) (p  ε1 )

log n 2(pε ) 1

,

 N , from (12) we can find some constant C13  C 

C9 v n

n  1(1δ) (pε )δ log n  1pε 1

1



n (1δ) (p  ε1 ) log n 

 ε  1 , we have 1

 PA n 3

It

C10 v n

n (1δ) (p ε1 )δ log n 

p ε1

,

n

 Ac

n 1

for some C9>0 and C10>0. Hence

follows



   

p  ε1

 C

Similarly α n

p  ε1

C11 C C  1   2(1δ) (p ε ) 12  (1δ) (p ε1 ) p  ε1 1 α  n 2n log n  log n 2(pε1 ) C13

Since p

We have

y n1 

For n largesay n

P An  Ac

y n 1



  

such that,

Consequently for n large,

n 1

log n  1

p  ε1

C12

C7 C  1   182α  o 1 2α  , where C7  0 and 1 α x x x  C8  0 are constants. Hence for x large, one can find C  0 such that,



p  ε1

Similarly we get

y n 2α  y n 2α1 ~

P An  Ac

  

C 9

p  ε1

C10 n (1δ) (p  ε1 )  δ log n 

Since f is the density function of a non-negative stable r.v. Density of stable law is given by, f(x)

log n  1

p  ε1

C11 n δ

~

and

n  1 vn



y n 1

(1 δ) (p  ε1 )  δ α

C9

(1 δ) (p  ε1 )  δ

that



13

n n 3

(1 δ) (p  ε1 )

 PA n 3

n

 Ac

1  . log n pε1

n 1

 

and

hence

P A n  A c i.o.  0 . n 1

Which implies the proof of (7) and (5) by Lemma 2 and consequently proof of (1) follows from (5).

To prove (8), we need to prove that, for some d>0,

International Educational Scientific Research Journal [IESRJ]

32

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

(1 δ) (p  ε1 )  δ p  ε1   α  P M u  C 4 n log n  α i.o.  d  0. n  

(13)

Using Lemma 1 (Extended Borel-Cantelli Lemma) and the Hewitt-Savage zeroone law, (2) will be proved.

Define

n k  [k θ ], 0  θ  1

and let

n C P M u  y n  ~ k 15  n k  k  yn k 2n k ~

Dk  (1 δ) (p  ε1 )  δ p  ε1  α log n k  α  M u n C 4 n k k  (1 δ) (p  ε1 )  δ p  ε1 α log n k  α   2C 4 n k 

nk

 C15

(1 δ) (p  ε1 )  δ

C

nk

log n k 

16 -(1 δ) (1 ε1 - p)

log n k pε

p  ε1 α

1

Therefore (1 δ) (1 ε - p)

1 n P M u  y n   C16 k p nk k   log n k  ε1

Then,

(15)

where C 16 is some positive constant.

PD k  

Similarly following above process, we can find some constant

 log n k   P M u  2C 4 n k nk   (1 δ) (p  ε1 )  δ p  ε1   α   P M u  C 4 n k log n k  α  nk   (1 δ) (p  ε1 )  δ α

p  ε1 α

(1 δ) (1 ε1 - p) yn   n P M u  k   C17 k nk  2  log n k pε1 

(16)

Substituting (15) and (16) in (14), we can find some constant k2(>0) such that, for all

 P M u  2C4 n k nk   P M u  y n  nk k  

  

PD k   C18 where

y n  2C 4 n k

Where

log n k 

p ε1 α

(1 δ) (p  ε1 )  δ α

log n k 

k

p  ε1 α .

In view of (9), we can observe that, condition (2) of Heyde (1967) is satisfied by

yn

k

can show that,

yn

Also since

nk

  0 as p

k .

1  limsup n 

we can find some constants

nk

yn

 1  ε 

1 α

, by Theorem in Hyede(1967),

k

C14 ( 0) and k1such that, for all k  k 1 ,



P M u  y n  ~ C14 P X1  y n nk k  k  Again using (9), we can find some constant

kk

2, (1 δ) (1 ε1 - p)

log n k pε

, 1

 PDk   C18 

nk

(1 δ) (1 ε1 - p)

log n k 

p  ε1

k (1δ) (1ε1 -p) p  ε1 k  k 2 log k 

 C19 

k k2

k k2

For some constant

C19 ( C18 ) and hence,

,

(17)

 PD    .

k k 2

k

s  (log k) η , η  1 , we have PD k  D s 

Let

k

yu

nk

C18 ( 0) and

n k  [k θ ], 0  θ  1 . This implies that,

. Again following proof of Lemma C in Vasudeva and Divanji (1991), we

Mu

such

(14)

Consider, (1δ) (p  ε1 )  δ α

C17 ( 0)

that



.

C15 ( 0) such that

(1 δ) (p  ε1 )  δ p  ε1  α log n k  α  M u n  P C 4 n k k 

 2C 4 n k C4ns

(1 δ) (p  ε1 )  δ α

 2C 4 n s  PD k 

(1 δ) (p  ε1 )  δ α

log n k 

log n s 

(1 δ) (p  ε1 )  δ α

p  ε1 α

log n s 

p  ε1 α

,

 Mu

p  ε1 α

ns

  

(1 δ) (p  ε1 )  δ p  ε1  α log n s  α P M u - M u  C 4 n s ns nk  (1 δ) (p  ε1 )  δ p  ε1 α log n k  α   2C 4 n k 

International Educational Scientific Research Journal [IESRJ]

33

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

Again following similar lines of (10) and Heyde’sTheorem , we have for

s  (log k) η , η  1 , PD k  D s 



 P D k  u n - u n s k

(1 δ) (p  ε1 )  δ p  ε1  α   P D k  P  M u  C 4 n s log n s  α ns 



Following similar lines of (10), we can find some constants

.

k4(>0) such that, for all

(1 δ) (p  ε1 )  δ p  ε1  α log n s  α  P  X1  C 4 n s  

C 20 ( 0)

and k3(>0) whereby,

PA n  , we can find for all k  k 3 and

s  (log k) , η  1 , η

(18)

(k  1)  s  (log k) η , η  1, we can note that, D k  D s 

Now for

(1 δ) (p  ε1 )  δ p  ε1  α log n k  α  M u n  C 4 n k k 

C4n s

(1 δ) (p  ε1 )  δ α

(1 δ) (p  ε1 )  δ α

 2C 4 n s

p  ε1 α

log n k 

log n s 

(1 δ) (p  ε1 )  δ α

p  ε1 α

p  ε1 α

Mu

ns

(1 δ) (p  ε1 )  δ α

 C4n s

log n k 

(1 δ) (p  ε1 )  δ α

p  ε1 α

k  k5 , (1 δ) (p  ε1 )  δ p  ε1   α  P  Mu  C4n s log n s  α  ns   C 22 ~ -(1δ) ε1 log n s pε1 ns

(1 δ) (p  ε1 )  δ p  ε1   α  P  Mu  C4n s log n s  α  ns  

  

θε (1 δ)

k 1  C 23 log k pε1

,

log n s 

Hence for all p  ε1 α

  

P D k  D s  

ns

Observe that

(1 δ) (p  ε1 )  δ α

 C4n s

Mu

nk

k  k6 ,

P D k  D s   C 23

θε (1 δ)

k 1 P D k . log k pε1

(19)

Note that,

P D k 

(1 δ) (p  ε1 )  δ p  ε1  α log n k  α  M u n P C 4 n k k 

Mu

C 23 ( 0) and k6(>0)

k  k6 ,

ns

Which implies

 2C 4 n k

C 22 ( C 21 ) and k5(>k4) such that, for all

such that, for all

(1 δ) (p  ε1 )  δ p  ε1  α log n k  α  M u n  C 4 n k k 

 2C 4 n k

p  ε1    P  Mu  C4n s log n s  α  ns   .

(1 δ) (p  ε1 )  δ p  ε1   α log n s  α . ~ C 21 n s P  X 1  C 4 n s  

Using the fact that s ≥ k+1, one can find some constants

,

 Mu

log n s 

C 21 ( 0) and

k  k4 ,

Using (9), we get some constants

PD k  D s   C 20 P D k P D s 

 2C 4 n k

 . 

(1 δ) (p  ε1 )  δ α

Applying the arguments used to get the upper bound of some constants

P D k  D s 

log n k 

(1 δ) (p  ε1 )  δ α

and

Mu

ns

(1 δ) (p  ε1 )  δ p  ε1   α   P  M u  2C 4 n k log n k  α  nk  

p  ε1 α

log n s 

p  ε1 α

 . 

are independent, we get,

International Educational Scientific Research Journal [IESRJ]

Again following steps similar to the above process of (19), we can find some constants C 24 ( 0) andk7(>0) such that, for all k  k 7 , we have θε (1 δ)

k 1 P D k   C 24 log k pε1

fors ≥ k+1.

(20)

34

Research Paper

E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016

From (19) and (20) there exists some constants

C 25 ( 0) and k8(>0) such

k  k8 ,

that, for all

n -1

 P D

k 1 s  (log k) 1

lim inf

2θ (1 δ)

k 1 P D k  D s   C 25 log k 2(p ε1 )

n

2 n 

k

 Ds 

 n    P ( Dk )   k 1 

.

2

 2C 20  0.

(26)

Now η n 1 (log k)

 k 1

Using (25) and (26) in (23) we have,

2θθ (1 δ)

n 1

(log k) η k 1   P D  D  C  k s 25  log k 2(p ε1 ) s  k 1 k 1

n

 P D lim inf

2θθ (1 δ)

n 1

n 

k 1  C 25  2(p  ε1 )  η k 1 log k  This implies that for n ≥ N1, we have n 1 (log k)

η

  P D k 1 s  k 1

(21)

From (17) we have for n ≥ N2, 

 P(D k 1

k

k (1δ)ε1 ε  C 26 log n  1 p ε1 log k 

)  C19

n

n

k 1 s 1

 Ds 

k

 n    P ( Dk )   k 1 

2

n -1



2

n

 P D

k 1 s  k 1

 D s    P ( Dk ) k 1

 n    P ( Dk )   k 1 

n -1

2

n

 P D

k 1 s  k 1

k

2

 n -1 (log k)



2

k   . Hence

 Ds 

 n    P ( Dk )   k 1 

 P D k  D s 

k 1 s  k 1

 n    P ( Dk )   k 1 

2

n -1



2

n

 P D

k 1 s  (log k) 1

k

 n    P ( Dk )   k 1 

2

 C 27  0.

1.

Chover, J. (1966). A law of iterated logarithm for stable summands, Proc. Amer.

Math. Soc.17,441-443.

2.

GootyDivanji (2004). Law of iterated logarithm for subsequences of partial sums which are in the domain of partial attraction of semi stable law, Probability and Mathematical Statistics, Vol.24, Fasc. 2,41, 433-442.

3.

GootyDivanji and K.N.Raviprakash (2016).A log log law for delayed random sums, preprint.

4.

Heyde, C.C. (1967). On large deviation problems for sums of random variables which are not attracted to the normal law, Ann. Math. Statist. 38(5):1575-1578.

5.

Lai, T.L. (1973). Limit theorems for delayed sums. Ann.Probab.2,432-440.

6.

Nielsen, O.B. (1961). On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables, Math.Scand 9:383-394.

7.

Spitzer, F (1964). Principles of random walk, Van Nostrand: Princeton, New Jercey.

8.

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

9.

Vasudeva, R and Divanji, G (1993). The law of the iterated logarithm for delayed sums under a nonidentically distributed setup, Theory Probab. Appl., 37(3), 497506.

(23)

2

By (22), the second term of the right side, tends to zero as consider,

 Ds 

REFERENCES (22)

n

k

k

ACKNOWLEDGEMENT The second author expresses his sincere gratitude to Professor M. Sreehari, 6-B, Vrundavan Park, New Sama Road, Vadodara-390 024, Gujarath-India, for suggesting the new concept "delayed random sums".

for some C26>0. Note that,

 P D

k 1 s 1

In view of (17) and (26), appealing to Lemma 1(Extended Borel-Cantelli Lemma) and Hewit-Savege zero-one law, we get P(Dki.o.)=1. Hence the proof of the Theorem is completed.

 D s   C 25 (log n) (12(p ε1 )η)

k

n

 Ds 

 n    P ( Dk )   k 1 

(24) 2

By (21) and (22), we have,  n -1 (log k)

2 lim n 

 P D

k 1 s  k 1

k

 Ds 

 n    P ( Dk )   k 1 

2

 0.

(25)

And by (18) and (22), we get that,

International Educational Scientific Research Journal [IESRJ]

35