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Abstract
In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.
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Asymont IM, Korshunov D. Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$. J THEOR PROBAB 2019. [DOI: 10.1007/s10959-019-00937-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
Abstract
Abstract
For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$
(
S
n
)
n
≥
0
in $${\mathbb {Z}}^d$$
Z
d
, $$d\ge 1$$
d
≥
1
, we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$
∑
x
∈
Z
d
f
(
l
(
n
,
x
)
)
of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$
l
(
n
,
x
)
=
∑
i
=
0
n
I
{
S
i
=
x
}
. Particular cases are the number of
visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$
f
(
i
)
=
I
{
i
≥
1
}
;
$$\alpha $$
α
-fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$
f
(
i
)
=
i
α
;
sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$
f
(
i
)
=
I
{
i
=
j
}
.
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Laurent C. Large deviations for self-intersection local times in subcritical dimensions. ELECTRON J PROBAB 2012. [DOI: 10.1214/ejp.v17-1874] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Becker M, König W. Self-intersection local times of random walks: exponential moments in subcritical dimensions. Probab Theory Relat Fields 2011. [DOI: 10.1007/s00440-011-0377-0] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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