Scaling limits of random walk bridges conditioned to avoid a finite set

Abstract

This paper concerns a scaling limit of a one-dimensional random walk $S^x_n$ started from $x$ on the integer lattice conditioned to avoid a non-empty finite set $A$, the random walk being assumed to be irreducible and have zero mean. Suppose the variance $\sigma^2$ of the increment law is finite. Given positive constants $b$, $c$ and $T$ we consider the scaled process $S^{b_N}_{[tN]}/\sigma\sqrt N$, $0\leq t \leq T$ started from a point $b_N \approx b\sqrt N$ conditioned to arrive at another point $\approx -c\sqrt N$ at $t=T$ and avoid $A$ in between and discuss the functional limit of it as $N\to\infty$. We show that it converges in law to a continuous process if $E[|S_1|^3; S_1<0] <\infty$. If $E[|S_1|^3; S_1<0] =\infty$ we suppose $P[S_1<u]$ to vary regularly as $u\to -\infty$ with exponent $-\beta$, $2\leq \beta\leq 3$ and show that it converges to a process which has one downward jump that clears the origin if $\beta<3$; in case $\beta=3$ there arises the same limit process as in case $E[|S_1|^3; S_1<0] <\infty$. In case $\sigma^2=\infty$ we consider the special case when $S_1$ belongs to the domain of attraction of a stable law of index $1<\alpha <2$ having no negative jumps and obtain analogous results.