LIMIT INFIMUM RESULTS FOR SUBSEQUENCES OF DELAYED RANDOM SUMS AND RELATED BOUNDARY CROSSING PROBLEM by International Educational Scientific Research Journal

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

LIMIT INFIMUM RESULTS FOR SUBSEQUENCES OF DELAYED RANDOM SUMS AND RELATED BOUNDARY CROSSING PROBLEM * GOOTY DIVANJI | . K.N. RAVIPRAKASH 1

Department of Statistics, Manasagangothri, University of Mysore, Mysuru – 570 006 – INDIA.

(*Corresponding Author). ABSTRACT





Let X n , n  1 be a sequence of i.i.d. strictly positive stable random variables with exponent α , 0  α  1 . We study a nontrivial limit behavior of linearly normalized subsequences of delayed random sums and extended to boundary crossing problem. Keywords: Law of iterated logarithm, Delayed random sums, Domain of attraction, Stable law.

Sn  , by normalizing in the power. Divanji and Vasudeva

1. INTRODUCTION Let X n , n  1 be a sequence of independent and identically distributed (i.i.d.) random variables (r.v.s.) and n

Sn   X k

let

n 1

Set

k 1

Ta  n

n an



i  n 1

Xi  Sn  a n  Sn , where a n ,  n  1 be a

non-decreasing function of a positive integers of n such

an ~ b n , where b n is n   n  log log n  . The non-increasing. Let γ n  log  an  that, 0  a n  n , for all n and

sequence

T



, n 1

Now parallel to the delayed sums Ta , we introduce n

MN  n

random n  Nn



j n 1

When variance is finite, Lai (1973) had studied the behavior of classical LIL for properly normalized sums Ta , at different values of a n ' s . He proved that these n

results are entirely different from LIL for partial sums . For independent but not identically distributed strictly positive stable r.v.s. Vasudeva and Divanji (1993) studied a non-trivial limit behavior of delayed sums Ta . n

is called a (forward) delayed

sum sequence (See Lai (1973)).

delayed

(1989) extended the same to the domain of partial attraction of a semi stable law with exponent α , 0  α  2 . When Gut (1986) established the classical LIL for geometrically fast increasing subsequences of Sn  . Torrang (1987) extended the same to random subsequences.

sums

as,

X j  Sn  Nn  Sn , where

N n , n  1 be a sequence of positive r.v.s. independent N of X n , n  1 such that Lim n  1 almost surely. n n

When X n ' s are i.i.d. symmetric stable r.v.s., Chover (1966) established the law of iterated logarithm (LIL) for

When r.v.s. are i.i.d. non-negative, which are in the domain of attraction of a completely asymmetric positive stable law, with exponent α , 0  α  1 , Gooty Divanji and Raviprakash (2016) studied Chover’s form of LIL for normalized delayed random sums

M



, n 1 .

Observations made by Gut (1986) motivated us to examine whether limit infimum for properly normalized

 

 

delayed random sums M N , k  1 can be studied, n k

for the i.i.d. strictly positive stable r.v.s. We answer in affirmative and extend the same to number of boundary crossings associated with this LIL.

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

Throughout the paper C, , , , ,  and k with or without a suffix or super suffix stand for positive constants with k and n confined to be positive integers. i.o. and a.s. stand for infinitely often and almost surely respectively.

r.v.s.

independent

 

 

sums M N , k  1 and in the last section, we extend n k

the study to boundary crossing problem.

2. SOME KNOWN RESULTS Lemma 2.1 (Extended Borel-Cantelli Lemma) Let E n  be a sequence of events with a common probability space  Ω, , P  .

n

Lim inf

 P E n   

and

n 1

(ii)

Lim inf n

P  E n i.o.  C .

k 1

s 1

 

P  E k Es 

 n   P E k      k 1 



C

then

function

0  α  1 . Then, as x  0 , α α 2 1-α   1-α   - x   x P  X1  x  ~ 2πα B α  exp - B  α   ,     B  α  = 1-α  α



πα    Cos  2  

M Nn



β Nnk





1α α

n  Nn

where

1 α-1

r.v.s, which are in the domain of attraction of a completely asymmetric positive stable law, with index α ,

0  α  1 and let N n , n  1 be a sequence of positive

International Educational Scientific Research Journal [IESRJ]

 1 a.s. .

 ε* a.s. ,



  log n k 

- ε1α-1

k  k0

 <  , 

for some k0 > 0, 1 α  nk  



 i 1

log

nk Nnk

 log log n

α 1  α  , k  

B  α   1  α  α

Lemma 2.3 Let X n , n  1 be a sequence of i.i.d. non-negative

 where ε = sup ε1 > 0: 

θ α  Bα 

For proof see Vasudeva and Divanji (1989).

X n , n  1 such that

β N nk  θα N

 iu α    πα   EexpiuX1   exp u 1  tan  , u  2      

M Nn

β Nn 

, k  1 be a subsequence of r.v.s independent of

Lemma 2.2 Let X1 be a positive stable r.v. with characteristic

α-1 α

k 

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

   1 a.s.

3 A NON-TRIVIAL LIMIT BEHAVIOR FOR LINEARLY DELAYED RANDOM SUMS Theorem 3.1 Let X n , n  1 be a sequence of i.i.d. strictly positive stable r.v.s. with indext α , 0  α  1 . Let n k  be a n integer subsequence such that Lim sup k  1 and let n k 1 k 

Then lim inf

-1

that

The proof follows similar lines of Theorem 2 of Gooty Divanji and K N Raviprakash(2016) and hence details are omitted.

N



If (i)

such

Lim Nnn = 1 a.s. Then, n

In the next section, we present some known results. In section 3, we established a non-trivial limit behavior of linearly normalized subsequences of delayed random

X n , n  1

α 1 α

 Cos

Xi  Sn

πα 2

k  Nn k



1 α-1

and

 Sn . k

Proof

Equivalently, we show that, for any ε  0 ,



 



 

P M N  ε *  ε β N n k i.o.  0 (1) nk   and

P M N  ε *  ε β N n k i.o.  1 (2) nk  

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

From the condition,

 1 a.s. as k   implies

that, for any δ  0,1, we have,

u n  N n  v n a.s., k

where,

(3)

u n  1  δ n k

and

v n  1  δ n k k

Consequently, one can notice that, as X n ' s are i.i.d. strictly positive stable r.v.s., we have, M u  M N  M v a.s. (4) nk



β n k 



 1+δ1  .

(5)

Using (4) and (5), it is enough to prove (1) and (2) as,





P M u  ε *  ε 1  δ1  βn k  i.o.  0 (6)  nk  where, M u



 Sn

k u n k

k  vn k



 C1θα  ε*  ε   log logn k  α . Taking x α 1





α 1 α

in Lemma 2.2, we can

  ε* ε  1δ1  β n k    ~ P  X1  1   α u n  k  α C2 exp   ε*  ε  α 1 log log n k 1  log log n k  2





 loglog n k 

1 2

  ε ε  logn k   *



α α 1

(ε*  ε) α 1



 Sn

1 α u nk

 log n k 

 Sn and

P M v  ε *  ε 1  δ1  βn k  i.o.  1(7)  nk  where, M v

 ε ε  1δ  β n

find some k1 (>0) and C2 ( > 0) such that,

as k   , which implies that there exits some δ1  0 , such that, β Nnk

exits some constant C1 > 0 such that,

as C1θα ε*  ε  log logn k 

 1 a.s. as

k   and (3), we can observe that, β N n k   1 a.s. β n k 

1  δ1  

 

In view of definition of β Nn k , we can notice that, there

By condition,

P  M u   ε*  ε  1  δ1  β  n k    nk  . *   ε ε  1δ1  β n k    = P  X1  1   α u nk  

 Sn

0  ε *  ε  ε * for some ε * sufficiently small and by the definition of ε and also by From the fact that

(8), we have, for some k1 ( > 0),

 P  M

k  k1

  ε*  ε 

nk 1 The condition lim sup n k 1 k 

 C2 

1 α *  ε ) α 1

   . 

k  k1  log n ( ε k

implies that, there exits some constant b>1 such that, n k 1  bn k , for some k large (8) From the fact that, X n ' s are strictly positive stable r.v.s. with exponent , 0  α  1 , we have

1  δ1  β  n k 

Mun 1 α

and X1 are

So that (6) follows by Borel - Cantelli Lemma. Consequently (1) follows from (6). To prove (7), we prove that, for some d > 0,

P  M v   ε*  ε  1  δ1  β  n k  i.o.   d  0.  nk 

u nk

(9)

identically distributed and hence, Define the events International Educational Scientific Research Journal [IESRJ]

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016





Now let s  t k , where

 * Dk   0  M vn  ε  ε 1  δ1  βn k  . k  

t k  k   log k  , for some  > 1. Then, P  D k Ds 

Then we have,

P  Dk   P  0  M v   ε*  ε  1  δ1  β  n k   . nk   From the fact that, X n ' s are strictly positive stable r.v.s. with exponent α , 0  α  1 , we have, M vn

1 v nα k

and X1 are identically distributed and hence,

  ε* ε  1δ1  β n k   P  Dk   P  X1   (10) 1  vnα  k  Again by the definition of β Nn k , we observe that, there

 

 0  M v   ε*  ε  1  δ1  β  n k   nk   P  0  M v   ε*  ε  1  δ1  β  n s   ns  

Observe that,

0  M v   ε*  ε  1  δ1  β  n k  ,    nk  * 0  M  ε  ε 1  δ β n  1   s    vn  s 

exits some constant C3>0 such that,

0  M v   ε*  ε  1  δ1  β  n k  ,    nk   * 0  M v  M v   ε  ε  1  δ1  β  n s    ns nk 

ε

and hence,

 ε  1  δ1  β  n k  1 α

vnk

 C3θ α  ε*  ε   log logn k 

α 1 α

. Again taking x as

C3θα  ε*  ε   log log n k  α in Lemma 2.2, we can find α 1

 ε*  ε  1  δ1  β  n k      P  D k   P X1  1 α   vnk   α C4 exp   ε*  ε  α - 1 log log n k 1  log logn k  2 C4





 log log n k 

1 2

α α-1

(11)

α α-1

0  ε *  ε  ε * for some ε * sufficiently small and by the definition of ε we have, From the fact that

P  0  M v   ε*  ε  1  δ1  β  n k    nk   k  k2  C4 

and hence PD k    .

Ds   P  Dk 

P  M v  M v   ε*  ε  1  δ1  β  n s   nk  ns 

(12) Again using the fact that, X n ' s are strictly positive stable r.v.s. with exponent α , 0  α<1 , we have,

M vn  M vn s 1 α

v ns  v n k

k 1 α

and

X1 are identically distributed and hence,

C4 ε ε  log n k  

P  Dk



ε ε  log n k  

we get,

some k2 ( > 0) and C4 ( > 0) such that,

P  D k Ds   0  M v   ε*  ε  1  δ1  β  n k  ,  nk .  P  0  M v  M v   ε*  ε  1  δ1  β  n s   ns nk   As M v and M v  M v are independent,

1 α

 *  εα - 1

P  M v  ns



  ε*  ε  1  δ1  β  n s      .  ε*  ε  1δ1  β n k    P  X1  1   α v  v  ns n k   

Following the steps similar to those used to get (11), we can find some constant C5(>0) and a k3(>0) and such that, for all k  k 3 ,



k  k2  log n k  ε

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P  M v  ns





E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

  ε*  ε  1  δ1  β  n s    . Notice C5

 log n s 

ε*  ε



α α-1

From the definition of ε * we have, ε *  1 , which implies

ε



 ε  1  1 and hence,

 ε ε  k *

α α-1

k k4

that, there exists some constant C6 (>C5) such that

P  M v  M v   ε*  ε  1  δ1  β  n s   nk  ns  . From  C6 P  Ds 

Therefore,

t k  k   log k  , for some  >1, observe that, M v and M v are independent and using the ns

k k4

k 1

s  k 1

P  Dk

k 1 s  k 1

   log k  PD k 

k 1

s  k 1

(14)

Using Lemma 2.2 in (10) and hence from (14), we can find some constant C7 (>0) such that,

k 1

s  k 1

  P D

 log k 

Ds   C 7  k 1

ε ε  log n k   *

α α-1

(15) From (8) we have, n k  b n 0 , where b>1 and n0 is fixed and hence, k

  k 1

s  k 1

P  Dk

log k

Ds   C7  k 1

ε ε  *

α α-1

(16)

Using (8) in (11), we can find some constant, C8 (>0) and a k4 (>0) such that, for all k  k 4 , n



k k4

P  Dk   C8 

k k4

 ε ε  *

α α-1

ε ε  k *

α α-1

International Educational Scientific Research Journal [IESRJ]

(17)

log n



P  Dk

Ds 

ε ε  *

 n 1 C α *  5 k 1  ε  ε  α - 1  k C  7  C9 ( 0), C5

k 1

k k4

 n  n  P D   k   P  D k     k 1  k 1  n log k C7  α

  PD k  D s     PD k 

k 1 s  k 1

Ds 

 

PD k  D s   PD k  which implies, n

1  C8  k k4 k

 n    P  Dk    k 1 

inequality s  k  1 we have,

k k4 1

 k.

From (16) and (17), we have, there exists C9 (>0) such that,

 

Now for k  1  s  t k , where



α α-1

(13)

for some s  t k , where, t k  k   log k  for some  >1.

 ε ε  *

 P D   C 

(12), we have,

PD k  D s   C 6 PD k PD s  ,



 k or

k  1  s  t k ,

it holds for some  >1.



α-1

  log n   (18)

t k  k   log k  , for

From (13), we have, s  t k ,

t k  k   log k  , for some  >1, n

k 1

stk

  P D

Ds 

k n

 C6  k 1

 P D  P D 

stk

 n  n   C6   P  D k    P  D s    k 1  s 1    

 n   C6   P  D k    k 1   

 n   n    P  D k    C6   P  D k    k 1   k 1 

(19) 11

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

X n , n  1

Observe that, n

  PD k  D s  ~ n

Lim inf

k 1 s  t k

k 

 PD k  Ds  ~

2

k 1 s  k 1

n -1  t k  n 2    PD k  D s   PD k  D s  ~   k 1  s  k 1 s  t k 1 

2



k 1 s  k 1

P  Dk n-1

 2



P Dk

Ds 

. (20)



Then

 1 a.s. ,

k 1 s 1

 n    PD k   k 1 

2   PD k  Ds  k 1 s  k 1





(21)





(22)

P M N  1  ε  β N n k i.o.  0 nk   P M N  1  ε  β N n k i.o.  1 nk  

n -1

2   PD k  Ds 

 n    PD k   k 1 

Proof: To prove the theorem, it is enough to show that, for any ε  0,1 ,

P M u  1  ε 1  δ1  βn k  i.o.  0 (23)  nk  where M u  S n  u  S n and

  PD k  D s 



k 1 s  t k 1

 n    PD k   k 1 

k 1 s 1

 C10  0 .

k k 2

to E B C Lemma 2.1 and Hewitt - Sevage zero-one law, proof of (7) follows from PD k i.o.  1 and consequently proof of (2) follows from (7). Hence proof of the theorem is completed.

P M v  1  ε 1  δ1  βn k  i.o.  1 (24)  nk  where M v  S n  v  S n k

k 

    PD k   k 1  In view of the series  PD k    and (19), appealing n

The condition Lim inf

  PD k  D s 

From (18), (19) and (20), one can find some constant C10 (> 0) such that, n

 1 a.s.

We follow similar steps of Theorem 3.1, it is sufficient to prove (21) and (22) as,

This implies,

n -1



β Nnk

and

Ds 

k 1 s  t k 1

M Nn

Nnk

that

n-1

such

nk  0 implies that, there exits n k 1

some ρ  0 such that,

n k 1 

nk , for k large ρ

(25)

 

This in turn implies that, M v

, k  1  is a sequence of 

mutually independent r.v.s. To prove (23), it is sufficient to show (6) for ε  1 . For any ε  0,1 , by Lemma 2.3, we claim that, *

Lim inf k 

M vn

β n k 

 Lim inf k 

M Nn



β Nnk



 1 a.s. which establishes

(23), for ε  1 and consequently proof of (21) follows *

Theorem 3.2 Let X n , n  1 be a sequence of i.i.d. strictly positive stable r.v.s. with index α , 0  α  1 . Let n k , k  1 be a integer subsequence such that Lim inf

N



k 

nk n k 1

 0 and let

from (23), for ε  1 . *

Now to establish (24), for ε  1 , we proceed as in Allan Gut (1986). Define, *

,k  1 be a sequence of r.v.s . independent of

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

m j  min k : n k  M j  , where j=1,2…. and M is

Define N   ε  

k 1

chosen such that

1  1 . Hence using (25), we get ρ M

a proper r.v. in view of (26).

that, M j  n m 

Mj and ρ2

Let

ρ  2

j-1

1  2 , j=1,2….; Consequently,  n m   j ρ M

satisfies the condition Lim sup j

with

  log n

k  k0



n m j-1 nm j

 1 of Theorem 2.1



 1 1

, for all ε  1 (i.e. ε *  1) .

Hence (24) follows from Theorem 3.1, for ε  1 . Consequently (22) follows from (24) and hence proof of Theorem is completed. *

4. NUMBER OF BOUNDARY CROSSING PROBLEM In this section, we study some boundary crossings r.v.s. related to Theorems 3.1 and 3.2. Define for any ε  0 ,

1, Yk  ε  =  0,

if M N



  θ ± ε  β N nk





 ε 

be the corresponding sequence of partial

sums. i.e. N m ε   k

 Yk ε  , where m k , k  1

is a

k 1

subsequence of sequence. If 

N   ε    Yk  ε  , we know that PN  ε     1 k 1

or N ε  is a proper r.v. of number of boundary crossings and hence it is interesting to study the existence of moments for this boundary crossing r.v.s. and obtain moments of this proper r.v. N ε  as Corollary to Theorems 3.1 and 3.2. This boundary crossing r.v.s. was studied by various authors like Slivka(1969) and Slivka and Savero (1970).

Corollary 4.1 For ε  0 and for any η, 0  η  1 , 

nk

η1

k 1



ENη <  , if



P M N  θ  ε  β N n k    . nk  

if  n k  is atleast geometrically fast

First we show that for η  1 , EN    and then claim that, the existence of lower moments follows from that of the higher moments.

if  n k  is atmost geometrically fast

Observe that

n  Nn

N

Proof

otherwise

where,

ε * ,  θ=  1,

and observe that N ε  is

 Y ε

 i 1

Xi  Sn

k  Nn

 Sn k

      -1    ε * ε  sup ε1  0:   log n k  1   , k k0       for some k 0  0 . For any ε  0 , we have  

from (1), P M N



 

 ε *  ε β N n k i.o.  0  (26)

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



EN   ε  =  P  M N   θ ± ε  β N n k  nk  k1 

Following similar arguments of the proofs of (1) and (21), we can find some constant C1 > 0 and k1 > 0 such that, for all k  k1 ,

EN  ε   C1 

 log n k  EN  ε  <  for k1

θ yields,

1  θ  

ε  2 

and by the definition of

η  1.

Consequently, we have EN <  for 0  η  1 . Thus the proof of the Corollary is completed. η

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E-ISSN No : 2455-295X | Volume : 2 | Issue : 9 | Sep 2016

REFERENCES [1] Chover, J. (1966) A law of iterated logarithm for stable summands, Proc. Amer. Math. Soc.17, 441 - 443. [2] Gooty Divanji and Vasudeva, R. (1989): Tail behavior of distributions in the domain of partial attraction on some related iterated logarithm laws, Sankhya, Ser A 51 (2), 196-204. [3] Gooty Divanji and Raviprakash, K.N. (2016) A log log law for delayed random sums. communicated to Calcutta Statistical Association Bulletin. [4] Gut, A. (1986) Law of Iterated Logarithm for Subsequences. Probability and Mathematical Statistics, 7, 27-58. [5] Lai, T.L. (1973), Limit theorems for delayed sums. Ann.Probab.2, 432-440. [6] Slivka, J. (1969) On the LIL, Proceedings of the National Academy of Sciences of the United States of America, 63, 2389-291. [7] Slivka, J. and Savero, N.C. (1970) On the Strong Law of Large Numbers, Proceedings of the American Mathematical Society, 24, 729-734. [8] Spitzer, F. (1964) Principles of random walk, Van Nostrand: Princeton, New Jercey. [9] Torrang, I. (1987) Law of the iterated logarithmCluster points of deterministic and random Subsequences. Prob.Math.Statist. 8, 133-141. [10] Vasudeva, R. and Divanji, G. (1988) Law of Iterated Logarithm for the Increments of Stable Subordinators. Stochastic Processes and Their Applications, 28, 293 â&#x20AC;&#x201C; 300. [11] Vasudeva, R. and Divanji, G. (1993) The law of the iterated logarithm for delayed sums under a non identically distributed setup. Theory Probab. Appl. 37(3):497-506.

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