Line crossing problem for biased monotonic random walks in the plane
Abstract
Description
In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=αx+d$, where $α,d \geq 0$. A $β$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(β+ 1)$ and $β/(β+ 1)$, respectively. Among our results, we show that if $β\geq \lceil α\rceil$, then the $β$-biased monotonic random walk, starting from the origin, crosses the line $y=αx+d$ for all $d\geq 0$ with probability 1.
11 pages
11 pages