The Annals of Probability

Weak limits of perturbed random walks and the equation $Y_t = B_t + \alpha\sup\{Y_s\colon s \leq t\}+\beta\inf\{Y\sb s\colon s \leq t\}$

Burgess Davis

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Let $\alpha$ and $\beta$ be real numbers and $f \in C_0 [0, \infty)$. We study the existence and uniqueness of solutions g of the equation $g(t) = f(t) + \alpha \sup_{0 \leq s \leq t} g(s) + \beta \inf_{0 \leq s \leq t} g(s)$. Carmona, Petit, Le Gall, and Yor have shown existence or nonexistence and uniqueness for some $\alpha, \beta$. We settle the remaining cases. We study the nearest neighbor walk on the integers, which behaves just like fair random walk unless one neighbor has been visited and the other has not, when it jumps to the unvisited neighbor with probability p. If $p < 2/3$, we show these processes, scaled, converge to the solution of the equation above for Brownian paths, with $\alpha = \beta = (2p - 1)/p$.

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Ann. Probab., Volume 24, Number 4 (1996), 2007-2023.

First available in Project Euclid: 6 January 2003

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60J15 60J65: Brownian motion [See also 58J65] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Reinforced random walk perturbed Brownian motion weak convergence


Davis, Burgess. Weak limits of perturbed random walks and the equation $Y_t = B_t + \alpha\sup\{Y_s\colon s \leq t\}+\beta\inf\{Y\sb s\colon s \leq t\}$. Ann. Probab. 24 (1996), no. 4, 2007--2023. doi:10.1214/aop/1041903215.

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