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

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.