An extension of a theorem on gambling systems (Q1907813)

Let \(\{X_n, n \geq 1\}\) be a sequence of \(S = \{0,1\}\)-valued random variables with the joint distribution \[ P[X_1 = x_1, \dots, X_n = x_n] = p(x_1, \dots, x_n) > 0, \quad x_i \in S,\;1 \leq i \leq n, \] so that \(P[X_i = x] = p_i^x (1 - p)^x\) for some \(0 < p_i < 1\), \(x \in S\), \(1 \leq i \leq n\), and let \(Y_1 = 1\), \(Y_{n + 1} = f_n (X_1, \dots, X_n)\) for \(n > 1\), where \(f_n : S^n \to S\) are any functions. Assume in addition that \(Z_n = \sum^n_{i = 1} Y_i \to \infty\) a.e. Limiting properties as \(n \to \infty\) of the relative frequency \(\sum^n_{i = 1} X_i Y_i/Z_n\) of occurrence of 1 up to time \(n\) are investigated by means of the likelihood ratio \[ r_n (\omega) = \left. \left[ \prod^n_{i = 1} p_i^{X_i} (1 - p_i)^{1 - X_i} \right] \right/p (X_1, \dots, X_n). \] Namely, it is shown that \[ \lim_n \left\{ \sum^n_{i = 1} (X_i - p_i) Y_i/Z_n \right\} = 0 \quad \text{a.e.} \] for \(\omega \in D = \{\omega : \liminf_n [r_n (\omega)]^{1/Z_n} \geq 1, \sum^\infty_{i = 1} Y_i = \infty\}\). Some special cases are also discussed and as a consequence the theorem on gambling system is deduced.