A strong limit theorem under no assumption of independence, stationarity or various dependences
The following theorem is proved. Let \((q_ n, n \geq 0)\) be a sequence of positive integers, \(I_ n = \{0,1, \dots, q_ n\}\), \((X_ n, n \geq 0)\) be a sequence of random variables, where \(X_ n\) takes on values in \(I_ n\), and \[ P(X_ 0 = x_ 0, \dots, X_ n = x_ n) = p(x_ 0, \dots, x_ n) > 0 \quad \text{for all} \quad x_ i \in I_ i,\;0 \leq i \leq n. \] If \(\sum^ \infty_{n=1} q^ 2_ n/n^ 2 < \infty\), then \[ \lim_ n {1 \over n} \sum^ n_{i=1} \biggl( X_ i - E \bigl( X_ i \mid X_ 0, \dots, X_{i-1} \bigr) \biggr) = 0 \quad \text{a.s.}. \]
scientific article; zbMATH DE number 681890