Total positivity, absorption probabilities and applications

A function K (x, y) of two real variables ranging over linearly ordered sets X and Y respectively is said to be totally positive of order r (TPrTypically, X is an interval of the real line, or a countable set of discrete values on the real line such as the set of all integers or the set of non-negative integers ; similarly for Y. When X or Y is a set of integers, we may use the term "sequence" rather than "function. "A related concept is that of sign regularity. A function K (x, y) is sign regular of order r if for every xi<x2< ■■■ < xm, yt <y2< ■•• < ym (x¡ eX; y¡ e Y) and 1 < m < r (0. 2) depends on m alone. In the case where em = (ᵢy<m-1V2 we tjjen sav tnat jî s "RRr, " where the letters stand for the abbreviation of sign reverse rule and suggest that (0. 2) would be positive except that the rows have been interchanged and appear in reverse order. If a TPr function K (x, y) is a probability density in one of the variables, say x, with respect to a a-finite measure p (x), for each fixed value of y, then