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