CRITERIONS OF THE SIMILARITY FOR RANDOM WALKS AND
BIRTH-AND-DEATH PROCESSES
Abstract: This paper is devoted to study the similarity of birth-and-death processes with a
discrete and continuous time. We discuss some relations between the measures of
orthogonality of the associated polynomials and the first return probabilities of two
-similar random walks and two -similar birth-and-death processes. We give the
necessary and sufficient conditions for -similarity of two random walks both in terms of
the corresponding spectral measures. We consider analogous conditions for -similarity of
two birth-and-death processes.
2000 AMS Mathematics Subject Classification: Primary 60J80; Secondary
42C05.
Key words and phrases: Associated polynomials, measures of orthogonality,
similar random walks, similar birth-and-death processes, first return probabilities.