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The distribution of the minimum observation until a stopping time, with an application to the minimal spacing in a Renewal process

Part of: Distribution theory Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  21 November 2025

Eutichia Vaggelatou
Eutichia Vaggelatou*
Affiliation:
National and Kapodistrian University of Athens
*
*Postal address: Department of Mathematics, National and Kapodistrian University of Athens, University Campus, Zografou 15784, Athens, Greece. Email: evagel@math.uoa.gr
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Abstract

Let $\{X_{i}\}_{i\geq1}$ be a sequence of independent and identically distributed random variables and $T\in\{1,2,\ldots\}$ a stopping time associated with this sequence. In this paper, the distribution of the minimum observation, $\min\{X_{1},X_{2},\ldots,X_{T}\}$, until the stopping time T is provided by proposing a methodology based on an appropriate change of the initial probability measure of the probability space to a truncated (shifted) one on the $X_{i}$. As an application of the aforementioned general result, the random variables $X_{1},X_{2},\ldots$ are considered to be the interarrival times (spacings) between successive appearances of events in a renewal counting process $\{Y_{t},t\geq0\}$, while the stopping time T is set to be the number of summands until the sum of the $X_{i}$ exceeds t for the first time, i.e. $T=Y_{t}+1$. Under this setup, the distribution of the minimal spacing, $D_{t}=\min\{X_{1},X_{2},\ldots,X_{Y_{t}+1}\}$, that starts in the interval [0, t] is investigated and a stochastic ordering relation for $D_{t}$ is obtained. In addition, bounds for the tail probability of $D_{t}$ are provided when the interarrival times have the increasing failure rate / decreasing failure rate property. In the special case of a Poisson process, an exact formula, as well as closed-form bounds and an asymptotic result, are derived for the tail probability of $D_{t}$. Furthermore, for renewal processes with Erlang and uniformly distributed interarrival times, exact and approximation formulae for the tail probability of $D_{t}$ are also proposed. Finally, numerical examples are presented to illustrate the aforementioned exact and asymptotic results, and practical applications are briefly discussed.

Keywords

Stopping timeminimum of random variablesrenewal processPoisson processexact distributionapproximating distributionincomplete gamma functionusual stochastic orderingIFR/DFR propertyexponential distributiongamma distributionuniform distributionreliability shock modelquality controlhypothesis testing

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62E15: Exact distribution theory
Secondary: 60E15: Inequalities; stochastic orderings 60G70: Extreme value theory; extremal processes

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Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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