Consider the class of even convex functions with [phi](0)=0 and concave derivative on (0,[infinity]). Given any [phi]-integrable martingale (Mn)n[greater-or-equal, slanted]0 with increments , n[greater-or-equal, slanted]1, the Topchii-Vatutin inequality (Theory Probab. Appl. 42 (1997) 17)...

Markov processes Xt on (X, FX) and Yt on (Y, FY) are said to be dual with respect to the function f(x, y) if Exf(Xt, y) = Eyf(x, Yt for all x [epsilon] X, y [epsilon] Y, t [greater-or-equal, slanted] 0. It is shown that this duality reverses the role of entrance and exit laws for the processes,...

Let T=(T1,T2,T3,...) be a sequence of real random variables. We investigate the following fixed point equation for distributions [mu]: W[congruent with][summation operator]j=1[infinity] TjWj, where W,W1,W2,... have distribution [mu] and T,W1,W2,... are independent. The corresponding functional...

We study in a systematic form the contractive behavior of the map S of distributions to distributions , Xi are independent r.v., L(Xi) = F. Further we show higher and exponential moments of the fixed point. Applications of this structure are given for (a) weighted branching processes, (b) the...

A new approach to the asymptotic normality of the multivariate linear rank statistics is provided along with the Berry-Esséen and the Prohorov distance estimates for the remainder term in the convergence to normality.