Random walks with jumps having finite variance

We study the (two-sided) exit time and position of a random walk outside boundaries which are regularly varying functions of smaller order at infinity than the square root. A natural domain of interest is those random walks which are attracted without centring to a normal law, or are relatively stable. These are shown to have ‘stable’ exit positions, in that the overshoot of the curved boundary is of smaller order of magnitude (in probability) than the boundary, as the boundary expands. Surprisingly, this remains true regardless of the shape of the boundary. Furthermore, within the same natural domain of interest, norming of the exit position by, for example, the square root of the exit time (in the finite-variance case), produces limiting distributions which are computable from corresponding functionals of Brownian motion. We give a functional limit theorem for attraction of normed sums to general infinitely divisible random variables, as a means of making this, and more general, computations. These kinds of theorems have applications in sequential analysis, for example.

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A note on the asymptotic properties of correlated random walks

Journal of Applied Probability ◽

10.2307/3213964 ◽

1985 ◽

Vol 22 (4) ◽

pp. 951-956 ◽

Cited By ~ 2

Author(s):

J. B. T. M. Roerdink

Keyword(s):

Asymptotic Behavior ◽

Random Walks ◽

Asymptotic Properties ◽

Simple Relation ◽

Finite Variance ◽

Correlated Random Walk ◽

Expected Number ◽

Similar Relation ◽

Correlated Random Walks ◽

Multistate Random Walks

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.

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Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

Siberian Mathematical Journal ◽

10.1007/s11202-005-0097-8 ◽

2005 ◽

Vol 46 (6) ◽

pp. 1020-1038

Author(s):

A. A. Borovkov

Keyword(s):

Asymptotic Analysis ◽

Random Walks ◽

Finite Variance

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A note on the asymptotic properties of correlated random walks

Journal of Applied Probability ◽

10.1017/s0021900200108204 ◽

1985 ◽

Vol 22 (04) ◽

pp. 951-956 ◽

Cited By ~ 2

Author(s):

J. B. T. M. Roerdink

Keyword(s):

Asymptotic Behavior ◽

Random Walks ◽

Asymptotic Properties ◽

Simple Relation ◽

Finite Variance ◽

Correlated Random Walk ◽

Expected Number ◽

Similar Relation ◽

Correlated Random Walks ◽

Multistate Random Walks

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.

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A note on random walks in multidimensional time

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064057 ◽

1986 ◽

Vol 99 (1) ◽

pp. 163-170 ◽

Cited By ~ 2

Author(s):

Janos Galambos ◽

Imre Kátai

Keyword(s):

Random Walks ◽

Recent Work ◽

Random Variables ◽

Finite Variance ◽

Image Position ◽

Positive Integers ◽

Independent And Identically Distributed

Let Kr denote the set of r-tuples n = (n1, n2, …, nr), r ≥ 1, where the components ni are positive integers. Let {X, Xn, n ∈ Kr} be a family of independent and identically distributed random variables with positive mean EX = μ < + ∞ and finite variance VX = σ2 < + ∞. In a recent work, M. Maejima and T. Mori [2] have shown that, if X is integer valued, aperiodic and E∣X∣3 < + ∞, then, for r = 2 or 3,wherethe summation being extended over all members j = (j1,j2, …, jr) of Kr that satisfy jt ≤ nt for all 1 ≤ t ≤ r.

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