0000000000924411
AUTHOR
Prem S. Puri
Hitting straight lines by compound Poisson process paths
In a recent article Mallows and Nair (1989,Ann. Inst. Statist. Math.,41, 1–8) determined the probability of intersectionP{X(t)=αt for somet≥0} between a compound Poisson process {X(t), t≥0} and a straight line through the origin. Using four different approaches (direct probabilistic, via differential equations and via Laplace transforms) we extend their results to obtain the probability of intersection between {X(t), t≥0} and arbitrary lines. Also, we display a relationship with the theory of Galton-Watson processes. Additional results concern the intersections with two (or more) parallel lines.
The linear birth and death process under the influence of independently occurring disasters
A population developing according to a time homogeneous linear birth and death process is subjected to an independently occurring random sequence of disasters. Using an embedded Galton-Watson process with random environments explicit results about the probability of extinction and the asymptotic behavior of the process are obtained.