On Distributional Properties of Perpetuities

The Annals of Applied Probability, 2010

We consider the tail behavior of random variables R which are solutions of the distributional equation R d = Q + M R, where (Q, M ) is independent of R and |M | ≤ 1.

Colloquium Mathematicum, 2010

Let (ξ k) and (η k) be infinite independent samples from different distributions. We prove a functional limit theorem for the maximum of a perturbed random walk max 0≤k≤n (ξ 1 +. .. + ξ k + η k+1) in a situation where its asymptotics is affected by both max 0≤k≤n (ξ 1 +. .. + ξ k) and max 1≤k≤n η k to a comparable extent. This solves an open problem that we learned from the paper "Renorming divergent perpetuities" by P. Hitczenko and J. Weso lowski.

Mathematische Nachrichten, 1987

Probability Theory and Related Fields, 1976

Canadian Journal of Statistics, 1989

Key words and phrases: Large deviations, laws of the iterated logarithm, diffusions with random coefficients.

Electronic Journal of Probability, 2021

Let (ξ 1 , η 1), (ξ 2 , η 2),. .. be independent identically distributed R 2-valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for convergent perpetuities k≥0 b ξ1+...+ξ k η k+1 as b → 1−. Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.

Journal of Applied Probability, 2014

Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ∼ D(a 0 ,. .. , a d), Pr(B = (0,. .. , 0, 1, 0,. .. , 0)) = a i /a with a = d i=0 a i , and Y ∼ β(1, a). Then, as proved by Sethuraman (1994), X ∼ X(1 − Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution B k (a 0 ,. .. , a d) with k an integer such that the above result holds when B follows B k (a 0 ,. .. , a d) and when Y ∼ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi-Bernoulli. Finally, the case where the integer k is replaced by a positive number c is considered when a 0 = • • • = a d = 1.

International Journal of Contemporary Mathematical Sciences, 2006

The class of ν-infinitely divisible (ID) distributions, which arise in connection with random summation, is a reach family including geometric infinitely divisible (GID) and geometric stable (GS) laws. We present two simple results connected with triangular arrays with random number of terms and their limiting ν-ID distributions as well as random sums with ν-ID distributed terms. These generalize and unify certain results scattered in the literature that concern the special cases of GID and GS laws.

2007