Finite-time trajectorial estimates for inhomogeneous random walks

TL;DR

This work develops non-asymptotic, finite-horizon bounds for time-inhomogeneous, integer-valued random walks with independent but non-identically distributed increments, uniformly tiltable within a compact interval. By combining an inhomogeneous local limit theorem with Doob-type submartingale techniques and a refined Gaussian-approximation framework, the authors derive sharp, uniform bounds for local limits, small-ball probabilities, positivity/exit events, and tail behavior, both for free-end trajectories and bridges. The contributions include a robust LLT in the Gaussian regime, uniform positivity estimates at fixed and variable endpoints, and detailed small-ball and excursion estimates, applicable to processes obtained through time-dependent tilting of increments. These results provide a versatile toolkit for finite-time analysis of constrained, time-inhomogeneous random walks and related stochastic models such as tilted or driven polymer-like systems. The framework emphasizes uniformity over admissible increment sequences and relaxes moment assumptions, broadening the scope of potential applications in probabilistic and statistical physics contexts.

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.

Figures (6)

• Figure 1: Construction in the proof of Lemma \ref{['lem:micro_small_ball_walk_LB']}: the path is forced to pass through the interval $[-\lambda/2,\lambda/2]$ at each time $L_1,\dots,L_{\ell-1}$.
• Figure 2: Construction in the proof of Lemma \ref{['lem:micro_small_ball_walk_UB']}: the path is forced to remain in the interval $[-\lambda,\lambda]$ and has to pass through the interval $[-\lambda,\lambda]$ at each time $L_1,\dots,L_{\ell}$, but is unconstrained otherwise.
• Figure 3: Upper bound in Lemma \ref{['lem:positivity_general_walks:moments']}: If the random walk trajectory reaches a height above $C\sqrt{n}$, then splitting the path at the first such point, as in the picture, shows that the final height is the sum of three terms of order at most $\sqrt{n}$.
• Figure 4: Construction in the proof of Lemma \ref{['lem:Gaussian_approx_walk']}: The increments between times $L_k$ and $L_{k+1}$ have to lie in the yellow region.
• Figure 5: Lemma 6.1: Theorem \ref{['thm:Z_bridge:small_ball']} can be used to estimate the contribution of the middle piece.

Theorems & Definitions (54)

• Lemma 1.1
• proof
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof