The distribution of the sojourn time for the Brownian excursion (Q1807738)

Let \(\{B(t),0<t<1\}\) be a standard Brownian bridge and let \(\{B^{*}(t),0<t<1\}\) be a standard Brownian excursion. Denote by \(w(\alpha)=\int_0^1 \delta(B(t))dt\) the sojourn time of \(B(t)\) spent in the set \((\alpha,\infty)\), \(\alpha>0\), in the time interval \((0,1)\) and use the notation \(w^{*}(\alpha)\) for the relevant sojourn time for the Brownian excursion \(B^{*}(t)\). The author utilizes the random walk approach to determine the distributions and moments of \(w^{*}(\alpha)\) and \(w(\alpha)\). He approximates the processes \(B(t)\) and \(B^{*}(t)\) by suitably chosen random walks and uses certain limit theorems to obtain the moments \(E\{[w^{*}(\alpha)]^r\}\) and \(E\{[w(\alpha)]^r\}\), \(r=1,2,\dots\), explicitly. These moments uniquely determine the distribution functions for corresponding sojourn times. Thus, explicit formulas which are suitable for numerical calculations are established and their applications to the problems connected with random trees, queueing and branching processes are presented.