Uniform Lorden-type bounds for overshoot moments for standard exponential families: small drift and an exponential correction
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arXiv
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| Natura: | Preprint |
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2026
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| author | Kalimulina, El'mira Yu. Kelbert, Mark Ya. |
| author_facet | Kalimulina, El'mira Yu. Kelbert, Mark Ya. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09504 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform Lorden-type bounds for overshoot moments for standard exponential families: small drift and an exponential correction Kalimulina, El'mira Yu. Kelbert, Mark Ya. Probability Statistics Theory 60K05 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. |
| title | Uniform Lorden-type bounds for overshoot moments for standard exponential families: small drift and an exponential correction |
| topic | Probability Statistics Theory 60K05 |
| url | https://arxiv.org/abs/2603.09504 |