Asymptotic behaviour of the time a receding level is attained by a random walk with delay. (English. Russian original) Zbl 0745.60071

Korolyuk, D. V.

Theory Probab. Math. Stat. 42, 73-76 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 64-67 (1990).

The paper considers a homogeneous Markov random walk on \([0,+\infty]\) with delay barrier at zero \[ S_{n+1}=(S_ n+\xi_{n+1})I(S_ n+\xi_{n+1}>0), \qquad n>0, \quad S_ 0\geq 0, \] where \(\xi_ i\), \(i\geq 1\), are independent identically distributed random variables, and \(I\) is the indicator of an event. Let \(\tau^ T_{S_ n}\) be the time of attainment of the domain \([T,+\infty)\) as \(T\to\infty\), \(f_ 0(x,T)=P\{\tau^ 0_ x>\tau^ T_ x\}\) called the probability of ruin. Two variants of random walks are investigated in which it is possible to obtain an asymptotic expression for the probability of ruin: 1. integer-valued lower semicontinuous random walk; 2. a walk with exponentially distributed negative jumps.

Reviewer: L.Lakatos (Budapest)

MSC:

60G50	Sums of independent random variables; random walks
60F99	Limit theorems in probability theory

Keywords:

random walk; time of attainment of the domain; probability of ruin; lower semicontinuous random walk; exponentially distributed negative jumps