On the distribution of the position of a randomly accelerated particle. (English) Zbl 0923.60059

Conti, Pier Luigi; Orsingher, Enzo

Theory Probab. Math. Stat. 56, 167-174 (1998) and Teor. Jmovirn. Mat. Stat. 56, 161-168 (1997).

The authors consider the process \(X(t)=A(0)\int_0^t(t-s)(-1)^{N(s)}ds\), where \(N(t)\) denotes the number of events of a homogeneous Poisson process in \([0,t],\) and the random variable (independent of \(N(t)\)) \(A(0)\) takes values \(\pm a\) with probability \(1/2\). This process is used for modelling a uniformly accelerated one-dimensional motion. The authors derive recurrent relationships for the conditional distributions \(P(X(t)\leq x\mid N(t)=n)\), \(n=1,2,\dots\) . The limit behaviour of such conditional distributions is investigated.

Reviewer: A.Ya.Olenko (Kyïv)

MSC:

60G70

Extreme value theory; extremal stochastic processes

Keywords:

order statistics; Poisson processes; random acceleration; telegraph signal; Dirac delta function