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

Let ᵢ≥1 be a sequence of independent and identically distributed random variables and T∈\1,2,…\ a stopping time associated with this sequence. In this paper, the distribution of the minimum observation, min₁,X₂,…,XT\ , 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 Xᵢ . As an application of the aforementioned general result, the random variables X₁,X₂,… are considered to be the interarrival times (spacings) between successive appearances of events in a renewal counting process ₜ,t≥0\ , while the stopping time T is set to be the number of summands until the sum of the Xᵢ exceeds t for the first time, i.e. T=Yₜ+1 . Under this setup, the distribution of the minimal spacing, Dₜ=min₁,X₂,…,X_Yₜ+1\ , that starts in the interval [0, t ] is investigated and a stochastic ordering relation for Dₜ is obtained. In addition, bounds for the tail probability of Dₜ 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 Dₜ . Furthermore, for renewal processes with Erlang and uniformly distributed interarrival times, exact and approximation formulae for the tail probability of Dₜ are also proposed. Finally, numerical examples are presented to illustrate the aforementioned exact and asymptotic results, and practical applications are briefly discussed.