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

El'mira Yu. Kalimulina, Mark Ya. Kelbert

Abstract

We study the overshoot \(R_b=S_{τ(b)}-b\) of a random walk with independent identically distributed increments from a standardised one-parameter exponential family, with primary emphasis on the small-drift regime \(θ\downarrow0\). Unlike the classical renewal-process setting with nonnegative increments, we allow sign-changing increments and assume only a positive drift \(μ_θ>0\). For each \(k\in\mathbb N\) we obtain Lorden-type moment bounds, uniform in the barrier \(b\), for \(\E_θ[R_b^k]\) 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 \(θ\in[0,θ^\ast]\). 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 \(θ\), and also uniformly over all \(b\ge0\) in the small-drift regime. Counterexamples are provided showing that the stronger inequality with \(kμ_θ\) in the denominator cannot hold uniformly in \((b,θ)\). 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.

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

Abstract

We study the overshoot \(R_b=S_{τ(b)}-b\) of a random walk with independent identically distributed increments from a standardised one-parameter exponential family, with primary emphasis on the small-drift regime . Unlike the classical renewal-process setting with nonnegative increments, we allow sign-changing increments and assume only a positive drift . For each we obtain Lorden-type moment bounds, uniform in the barrier , for with an explicit remainder term decaying exponentially in . 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 to the limiting random variable as , uniformly in . As a consequence, the classical constant \((k+2)/(k+1)\) arising in residual-life bounds improves to for sufficiently large at fixed , and also uniformly over all in the small-drift regime. Counterexamples are provided showing that the stronger inequality with in the denominator cannot hold uniformly in \((b,θ)\). Finally, the exponential CDF estimate is interpreted in terms of optimal transport: we obtain exponential convergence in the metric , 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.
Paper Structure (12 sections, 17 theorems, 80 equations)

This paper contains 12 sections, 17 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Overshoot moment bounds: statements and proofs of the main results
  3. The classical moment bound and its improvement in the large-barrier regime
  4. Overshoot bounds in coupling problems and estimates of the rate of convergence: Wasserstein distance, smoothed total variation and the error of replacing $R_b$ by $R_\infty$
  5. Examples of the use of overshoot bounds in applications
  6. Error control for Lipschitz functionals
  7. Smoothed total variation (fixed smoothing)
  8. Threshold stopping: explicit approximation error
  9. Conclusion
  10. Counterexamples: why the strengthened bound with $k\mu_\theta$ in the denominator is false in general: two counterexamples
  11. Counterexample 1: deterministic increments
  12. Counterexample 2: a standardised exponential family based on the uniform distribution

Key Result

Theorem 1

Assume that $X_1,X_2,\dots$ are nonnegative a. s. and Then, in the notation of eq:tau_def--eq:overshoot_def, for every $b\ge 0$ the inequality holds.

Theorems & Definitions (39)

  • Theorem 1: Lorden's inequality (Lorden, 1970)Lorden1970
  • Definition 1: Strict ascending ladder epochs and heights Feller71
  • Lemma 1: Wong2025
  • Definition 2: Renewal function for ladder heights
  • Proposition 1: Limiting distribution and limiting overshoot moments
  • Definition 3: Standard exponential family
  • Remark 1: Non-lattice property and applicability of the key renewal theorem
  • Proposition 2: Uniform exponential convergence of the overshoot Wong2025
  • Remark 2
  • Lemma 2
  • ...and 29 more