MathDB
sequence of random variables

Source: miklos schweitzer 2011 q10

August 29, 2021
probability and statsMiklos Schweitzer

Problem Statement

Let X0,ξi,j,ϵkX_0, \xi_{i, j}, \epsilon_k (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that ξi,j\xi_{i, j} (i, j ∈ N) have the same distribution, ϵk\epsilon_k (k ∈ N) also have the same distribution. E(ξ1,1)=1\mathbb{E}(\xi_{1,1})=1 , E(X0l)<∞\mathbb{E}(X_0^l)<\infty , E(ξ1,1l)<∞\mathbb{E}(\xi_{1,1}^l)<\infty , E(ϵ1l)<∞\mathbb{E}(\epsilon_1^l)<\infty for some l∈Nl\in\mathbb{N} Consider the random variable Xn:=ϵn+∑j=1Xn−1ξn,jX_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j} (n ∈ N) , where ∑j=10ξn,j:=0\sum_{j=1}^0 \xi_{n,j} :=0 Introduce the sequence Mn:=Xn−Xn−1−E(ϵn)M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n) (n ∈ N) Prove that there is a polynomial P of degree ≤l/2\leq l/2 such that E(Mnl)=Pl(n)\mathbb{E}(M_n^l) = P_l(n) (n ∈ N).
Back to ProblemsView on AoPS