Uniform Lorden-type bounds for overshoot moments for standard exponential families: small drift and an exponential correction

Abstract

We study the overshoot \(R_b=S_{\tau(b)}-b\) of a random walk with independent identically distributed increments from a standardised one-parameter exponential family, with primary emphasis on the small-drift regime \(\theta\downarrow0\). Unlike the classical renewal-process setting with nonnegative increments, we allow sign-changing increments and assume only a positive drift \(\mu_\theta>0\). For each \(k\in\mathbb N\) we obtain Lorden-type moment bounds, uniform in the barrier \(b\), for \(\E_\theta[R_b^k]\) with an explicit remainder term decaying exponentially in \(b\). The proof reduces the problem to the renewal process of strict ascending ladder heights and combines a simple bound for the limiting overshoot moments with a uniform exponential estimate for the rate of convergence of the distribution functions of \(R_b\) to the limiting random variable \(R_\infty\) as \(b\to\infty\), uniformly in \(\theta\in[0,\theta^\ast]\). As a consequence, the classical constant \((k+2)/(k+1)\) arising in residual-life bounds improves to \(C_k=1\) for sufficiently large \(b\) at fixed \(\theta\), and also uniformly over all \(b\ge0\) in the small-drift regime. Counterexamples are provided showing that the stronger inequality with \(k\mu_\theta\) in the denominator cannot hold uniformly in \((b,\theta)\). Finally, the exponential CDF estimate is interpreted in terms of optimal transport: we obtain exponential convergence in the metric \(W_1\), a quantile coupling with \(\E|\widetilde R_b-\widetilde R_\infty|=O(e^{-rb})\), error bounds for Lipschitz functionals and a total-variation bound for smoothed distributions.