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

Elmira Yu. Kalimulina, M. Ya. Kelbert|arXiv (Cornell University)|Mar 10, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper derives uniform Lorden-type bounds for overshoot moments R_b of a random walk with increments from a standard exponential family, in the small-drift regime, with an exponential correction in b and conditions under which the bound constant can be improved to 1 (C_k=1).

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 \(b o\infty\), 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.

Motivation & Objective

• Motivate and quantify overshoot moments for signed increments with positive drift in a standard exponential family.
• Develop Lorden-type bounds for E_theta[R_b^k] that are uniform in barrier b and theta in a small-drift regime.
• Reduce the classical bound constant from (k+2)/(k+1) to C_k=1 under large barrier or small drift conditions.
• Characterize the uniform rate of convergence to the limiting overshoot R_infty and connect to optimal transport metrics.
• Provide applications to threshold stopping and coupling via Wasserstein distance and smoothed total variation.

Proposed method

• Model increments X_i from a standard one-parameter exponential family with F_theta(dx)=e^{ heta x-\psi( heta)}F_0(dx).
• Reduce overshoot analysis to the renewal process of strict ascending ladder heights (H_n) and use the ladder-height renewal function U_theta^+(x).
• Use a key renewal theorem argument to establish an exponential convergence bound for P_theta(R_b≤y) to P_theta(R_infty≤y) uniform in theta.
• Derive an explicit bound for E_theta[R_b^k] with an exponential in b correction term: E_theta[R_b^k] ≤ (1/(k+1)) E_theta[(X_1^+)^{k+1}]/μ_theta + C (k Γ(k)/r^k) e^{-rb}.
• Show that for large b (Corollary 1) or small theta (Corollary 2) this bound reduces to E_theta[R_b^k] ≤ E_theta[(X_1^+)^{k+1}]/μ_theta.
• Interpret the exponential CDF estimate via Wasserstein distance (W1), quantile coupling, and smoothed total variation (TV) bounds.

Experimental results

Research questions

• RQ1 Can overshoot moments E_theta[R_b^k] be bounded uniformly in barrier b for increments from a standard exponential family with positive drift?
• RQ2 Does an exponential correction term allow tightening the classic Lorden bound, and under what regimes (large b or small drift) does the bound achieve C_k=1?
• RQ3 What is the rate of convergence of the overshoot distribution R_b to R_infty as b→∞, uniformly in theta, and how can this be quantified via optimal transport?
• RQ4 How can these bounds be applied to threshold stopping problems and coupling, including Lipschitz functionals and TV bounds after smoothing?

Key findings

• A uniform bound with exponential correction is proven: E_theta[R_b^k] ≤ (1/(k+1)) E_theta[(X_1^+)^{k+1}]/μ_theta + C (k Γ(k)/r^k) e^{-rb} for theta in (0, theta*].
• There exists a threshold b_0(θ,k) such that for b ≥ b_0(θ,k) the bound improves to E_theta[R_b^k] ≤ E_theta[(X_1^+)^{k+1}]/μ_theta (C_k=1).
• For sufficiently small drift θ, the improvement to C_k=1 holds uniformly over all b ≥ 0 (theorem corollaries).
• Proposition 2 provides a uniform exponential rate of convergence of the overshoot distribution: sup_theta≤θ* |P_theta(R_b ≤ y) − P_theta(R_∞ ≤ y)| ≤ C e^{-r(b+y)}.
• This leads to W1-convergence: sup_theta≤θ* W1(R_b, R_∞) ≤ (C/r) e^{-rb} and quantile coupling with E|R̃_b − R̃_∞| = O(e^{-rb}).
• Applications include Lipschitz functionals error bounds and smoothed TV bounds for overshoot distributions, and explicit error control for threshold stopping via E_theta[τ(b)].