On the range of random walk |
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Authors: | N Jain S Orey |
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Institution: | (1) Institute of Technology School of Mathematics, University of Minnesota, Minneapolis, Minnesota, U.S.A. |
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Abstract: | Let {S
n
, n=0, 1, 2, …} be a random walk (S
n
being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE
d
, thed-dimensional integer lattice. Letf
n
=Prob {S
1 ≠ 0, …,S
n
−1 ≠ 0,S
n
=0 |S
0=0}. The random walk is said to be transient if
and strongly transient if
. LetR
n
=cardinality of the set {S
0,S
1, …,S
n
}. It is shown that for a strongly transient random walk with p<1, the distribution of R
n
−np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S
0, …,S
n
}. For a finite setA inE
d
, let C(A=Σ
x∈A
) Prob {S
n
∉A, n≧1 |S
0=x} be the capacity ofA. A strong law forC{S
0, …,S
n
} is proved for a transient random walk, and some related questions are also considered.
This research was partially supported by the National Science Foundation. |
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Keywords: | |
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