Line crossing problem for biased monotonic random walks in the plane

Details Line crossing problem for biased monotonic random walks in the plane Line crossing problem for biased monotonic random walks in the plane

2007-09-20

arxiv.org/abs/0709.3316v1

Mathematics

Probability

11 pages

Scientific paper

In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$, where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and $\beta/(\beta + 1)$, respectively. Among our results, we show that if $\beta \geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random walk, starting from the origin, crosses the line $y=\alpha x+d$ for all $d\geq 0$ with probability 1.

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