In this paper, we show under weak assumptions that for R d = 1+M1+M1M2+. . ., where P(M ∈ [0, 1]) = 1 and Mi are independent copies of M , we have lnThe constant C is given explicitly and its value depends on the rate of convergence of ln P(M > 1 − 1/x). Random variable R satisfies the stochastic equation R d = 1 + M R with M and R independent, thus this result fits into the study of tails of iterated random equations, or more specifically, perpetuities.MSC 2010 subject classifications: Primary 60H25; secondary 60E99

We study solutions to the stochastic fixed point equation X d = AX + B when the coefficients are nonnegative and B is an "inverse exponential decay" (IED) random variable. We provide theorems on the left tail of X which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the IED class. We describe the lower envelope for these ARMA processes.

We prove the Lukacs-Olkin-Rubin theorem without invariance of the distribution of the "quotient," which was the key assumption in the original proof of (Olkin-Rubin in Ann Math Stat 33: [1272][1273][1274][1275][1276][1277][1278][1279][1280] 1962). Instead, we assume existence of strictly positive continuous densities of respective random variables. We consider the (cone variate) "quotient" for any division algorithm satisfying some natural conditions. For that purpose, a new proof of the Olkin-Baker functional equation on symmetric cones is given.

Abstract. We prove the Lukacs characterization of the Wishart distribution on non-octonion symmetric cones of rank greater than 2. We weaken the smoothness assumptions in the version of the Lukacs theorem of [Bobecka-Weso lowski, Studia Math. 152 (2002), 147-160]. The main tool is a new solution of the Olkin-Baker functional equation on symmetric cones, under the assumption of continuity of respective functions. It was possible thanks to the use of Gleason's theorem.

Abstract. We solve the logarithmic Cauchy functional equation in the symmetric cone with respect to two different multiplication algorithms. We impose no regularity assumptions on respective functions.Mathematics Subject Classification. Primary 39B52.