A strong limit theorem for generalized Cantor-like random sequences (Q1286673)

Let \(\{q_n, n\geq 0\}\) be a sequence of positive integers, \(I_n= \{0,1, \dots, q_n\}\). The sequence of random variables is called a generalized Cantor-like random sequence if for every \(n\), \(X_n\) takes on values in \(I_n\), and \[ P(X_0=x_0, \dots, X_n=x_n)= p(x_0,\dots, x_n)>0, \quad x_i\in I_i,\;0\leq i\leq n. \tag{1} \] The purpose of this paper is to give a strong limit theorem for these sequences: Let \(\{X_n, n\geq 0\}\) be a generalized Cantor-like random sequence with distribution (1). If there exists a constant \(\alpha\) such that \(q_n\leq\alpha n\) for all \(n\geq 1\), and if \[ \sum^\infty_{n=1} (1/n^2)D (X_n\mid X_0,\dots, X_{n-1})\text{ converges a.e.,} \tag{2} \] then \[ \lim_n(1/n) \sum^n_{i =1} \bigl[X_i-E(X_i \mid X_0, \dots,X_{i-1}) \bigr]=0\text{ a.e. }\tag{3} \] {}.