American Journal of Research in Humanities and Social Sciences 
ISSN (E): 2832-8019              Volume 18, | November, 2023                      

 

P a g e  | 8  www.americanjournal.org 

 

ON THE BEHAVIOUR OF SOME PROBABILISTIC CHARACTERISTICS OF 

THE OUTPUT OF MULTIDIMENSIONAL RANDOM WALK FROM 

EXPANDING SETS 
Gafurov M. U. 

Tashkent State Transport University 

mgafurov@rambler.ru 

 

A B S T R A C T K E Y W O R D S 

This paper establishes an analog of the well-known theorem of P. J. 

Bickel and J. A. Yahav on the number of exits of a multidimensional 

random walk from expanding sets. This theorem and related problems 

are carried forward for the moment of the first exit. 

 

Multidimensional 

random walk; expanding 

sets; moment of the first 

exit; number of exits. 

 

Introduction  

Let 𝑋1 … 𝑋𝑛, be independent identically distributed random variables (RV's) with values in 𝑅𝑑, 𝑑 ≥ 1. 

Define 𝑆0  =  0 and 𝑆𝑛 = ∑ 𝑋𝑖
𝑛
1  for n≥1. For any Borel set 𝐴 ⊂  𝑅𝑑 we set (formally) 

𝑁(𝐴) =  ∑ 𝐼 (𝑆𝑛

∞

𝑛=1

⋲ 𝐴),   𝑇(𝐴) = inf {𝑛, 𝑆𝑛 ∉ A}  

There are many references dealing with the study of the RV's N(A) and T(A) in the case d=1 (see, for 

example, the monograph [1]). In the general case, when d > 1, it was proved in (2] that if 

𝐸𝑁(𝐴) =  ∑ 𝑃(𝑆𝑛 ⋲ 𝐴)

∞

𝑛=1

< ∞ 

for any bounded set A, then 

𝐸𝑒𝑥𝑝 {𝑡 𝑁(𝐴)} <  ∞ 

for all |𝑡| ≤  𝑡0 where 𝑡0, is some positive number. The asymptotic behavior of the moments of N(A) 

for an expanding set A was investigated in the same paper. As far as the author knows, the distribution 

of T(A) when d > l has not yet been studied in depth. 

Our purpose is to determine the asymptotic behavior of the moments of T(A) on sets of the form 𝐴 =

 𝐴𝑥 = {𝑦 ⋲  𝑅𝑑, ||𝑦|| < 𝑥}, where ||·|| is any norm in 𝑅𝑑, as well as the behavior of the “first flight of 

stairs” 𝑆𝑇(𝐴𝑥) 𝑎𝑠 𝑥 → ∞, In this connection we have proved the following statements. 

THEOREM 1. If 𝐸 ||𝑋1|| < ∞ , then for all k ≥0 

lim
𝑥→∞

𝐸𝑇𝑘(𝐴𝑥)

𝑥𝑘
=

1

||𝐸𝑋1||𝑘
. 

This theorem complements a result in [2] on𝑁(𝐴𝑥). 



American Journal of Research in Humanities and Social Sciences 
Volume 17 Oct., 2023 

 

P a g e  | 9  www.americanjournal.org 
 

Let us consider a nondecreasing positive function φ(x) on [0, ∞) that is representable in the form 

𝜑(𝑥) = 𝑥𝑙𝐻(𝑥)  where 𝑙 ≥ 0 and H(x) is a slowly varying function in the sense of Karamata. 

THEOREM 2. Suppose that 𝐸𝑋1 ≠ 0 and 𝐸||𝑋1||
2

𝜑(||𝑋1||) < ∞, Then for any ε > 0 

∫ φ(𝑥)
∞

0

 𝑃{𝑆�̅�(𝐴𝑥) ⋲ 𝐴𝜀 𝑇(𝐴𝑥)
𝑐 } 𝑑𝑥 <  ∞ 

Here and in what follows 𝑆�̅� = ∑ (𝑋𝑖 − 𝐸𝑋𝑖)
𝑛
1 , and 𝐴𝑢

𝑐  is the complement of 𝐴𝑢. 

REMARK. By retracing the course of the proof of Theorem 2 it can be shown that 

(1) ∫ 𝜑(𝑥) 𝑃{𝑆�̅�(𝐴𝑥)
∞

0
⋲  𝐴𝜀 𝑥

𝑐 } 𝑑𝑥 < ∞  

It is easy to see that if ε→0, then the left-hand side of (1) converges to ∞, and the asymptotic behavior 

of the integral with respect to ε is of interest. 

Let B be the covariance matrix of the 𝑅𝑉 𝑋1. In this case we have the following assertion, which extends 

results in [3]. 

THEOREM 3. Suppose that 𝐸𝑋1 ≠ 0 and 𝐸||𝑋1||𝑡+2 < ∞. Then 

lim
𝜀→0

𝜀2(1+𝑙) ∫ 𝑥1𝑃{𝑆�̅�(𝐴𝑥)  ⋲   𝐴𝑢
𝑐

∞

0

} 𝑑𝑥 = ∫ 𝑥𝑙𝑃𝜂
∞

0

⋲ 𝐴
√𝑥
𝑐 } 𝑑𝑥 

where η is a normal RV with expectation the zero vector and covariance matrix ||𝐸𝑋1||−1𝐵. 

We mention some consequences of Theorem 3 when d = 1. 

COROLLARY 1. Suppose that 𝐸𝑋1 ≠ 0 and 𝐸𝑋1
𝑙+2 < ∞ then  

lim
𝜀→0

𝜀2(𝑙+2) ∫ 𝑥𝑙𝑃{|𝑆�̅�(𝐴𝑥) > 𝜀𝑥
∞

0

}𝑑𝑥 =  
2Г (𝑙 +

3
2)

√π(𝑙 + 1)
(

𝐷𝑋1

|𝐸𝑋1|
)𝑙+1 

 

COROLLARY 2. If 𝐸𝑋1 ≠ 0 and 𝐸𝑋1
2 < ∞ then 

lim
𝜀→0

𝜀2 {∫ 𝑥𝑙𝑃{|𝑆�̅�(𝐴𝑥) > 𝜀𝑥
∞

0

} − 𝑃{𝑆�̅�(𝐴𝑥)}]𝑑𝑥} = 0 

 

The following lemma, which is also of independent interest, can be used to prove the 

theorems given above. 

LEMMA. a) Suppose that 𝐸𝑋1 ≠ 0. Then for any 𝜀 > 0 

lim
𝑥→∞

𝑃 {|
𝑇(𝐴𝑥)

𝑥
− ||𝐸𝑋1||

−1
| > 𝜀} = 0 

b) If 𝐸𝑋1 = 0, then for sufficiently large C>0 

lim
𝑥→∞

𝑃|𝑇(𝐴𝑥) > 𝐶𝑥 = 1 

REMARK. By using the results in (4] it can be shown that the assertions of the lemma 

remain in force when the RV 𝑋𝑘 takes values in a separable Banach space. 

 

References 

1. Frank Spritzer, Principles of random walk, Van Nostrand, Princeton, N. J., 1964. 

2. P. J. Bickel and J. A. Yahav, Israel J. Math. 3 (1965), 181. 

3. M. U. Gafurov and S. H. Sirazdinov, Kybernetika (Prague) 15 (1979), 272 (Russian). 

4. T. A. Azlarov and N. A. Volodin, Limit Theorems, Random Processes and Their Applications, 

“Fan”, Tashkent, 1979, p. 15 (Russian). 

https://en.wikipedia.org/wiki/Not_equal_sign