scholarly journals Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments

Bernoulli ◽  
2007 ◽  
Vol 13 (1) ◽  
pp. 131-147 ◽  
Author(s):  
Sami Mustapha
Keyword(s):  
Random Walks ◽  
Spatially Inhomogeneous ◽  
Gambler’S Ruin

scholarly journals Spatially Inhomogeneous Populations with Seed-Banks: I. Duality, Existence and Clustering

2021 ◽  
Author(s):  
Frank den Hollander ◽  
Shubhamoy Nandan
Keyword(s):  
Random Walks ◽  
Seed Bank ◽  
Seed Banks ◽  
Model Parameters ◽  
Geographic Space ◽  
Spatially Inhomogeneous ◽  
And Migration ◽  
Coalescing Random Walks ◽  
Colony Migration

AbstractWe consider a system of interacting Moran models with seed-banks. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The colonies are labelled by $${\mathbb {Z}}^d$$ Z d , $$d \ge 1$$ d ≥ 1 , playing the role of a geographic space. The sizes of the active and the dormant population are finite and depend on the location of the colony. Migration is driven by a random walk transition kernel. Our goal is to study the equilibrium behaviour of the system as a function of the underlying model parameters. In the present paper, under a mild condition on the sizes of the active populations, the system is well defined and has a dual. The dual consists of a system of interacting coalescing random walks in an inhomogeneous environment that switch between an active state and a dormant state. We analyse the dichotomy of coexistence (= multi-type equilibria) versus clustering (= mono-type equilibria) and show that clustering occurs if and only if two random walks in the dual starting from arbitrary states eventually coalesce with probability one. The presence of the seed-bank enhances genetic diversity. In the dual this is reflected by the presence of time lapses during which the random walks are dormant and do not move.

Random walks with negative drift conditioned to stay positive

1974 ◽  
Vol 11 (4) ◽  
pp. 742-751 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Principal Result ◽  
Collective Risk ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Independent Identically Distributed

Let {Xk: k ≧ 1} be a sequence of independent, identically distributed random variables with EX1 = μ < 0. Form the random walk {Sn: n ≧ 0} by setting S0 = 0, Sn = X1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X1) that Sn, conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Random walks with negative drift conditioned to stay positive

1974 ◽  
Vol 11 (04) ◽  
pp. 742-751 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variable ◽  
Principal Result ◽  
Collective Risk ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Independent Identically Distributed

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ &lt; 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X 1) that Sn , conditioned on T &gt; n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ &lt; 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Non-homogeneous Random Walks

2016 ◽  
Author(s):  
Mikhail Menshikov ◽  
Serguei Popov ◽  
Andrew Wade
Keyword(s):  
Random Walks
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