Finite-time trajectorial estimates for inhomogeneous random walks

TL;DR

This paper develops finite-time, non-asymptotic estimates for inhomogeneous random walks, including local limit theorems and positivity probabilities, uniformly over classes of increment sequences, aiding analysis of time-dependent processes.

Contribution

It extends classical estimates to inhomogeneous random walks with non-identically distributed increments, providing uniform bounds over classes of sequences for finite time horizons.

Findings

Non-asymptotic local limit theorem established.

Uniform bounds on positivity probabilities derived.

Precise small-ball estimates for inhomogeneous walks obtained.

Abstract

We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on the final point and unconditional), and bounds on the probability that the random walk trajectory remains positive up to a given time (again, both conditional on the final point and unconditional). Two key features of this work are that the bounds are non-asymptotic, holding true for finite time horizons, and, crucially, that the latter hold uniformly over an entire class of admissible increment sequences. This provides a robust framework for applications. These results are, in particular, tailored for the analysis of processes derived through a time-dependent tilting of the increments of a time-homogeneous random walk.