Abstract
Let $X_1, X_2, \cdots$ be a sequence of random variables satisfying $E(X_{n + 1}\mid X_n, X_{n - 1}, \cdots, X_1) = a_1 X_n + a_2 X_{n - 1} + \cdots + X_{n - k - 1}, n \geqq k$, where $a_1 + a_2 + \cdots + a_k = 1$. Under certain general conditions, mainly that $\sup_nE|X_n| < \infty$, it is shown that $X_n - Y_n \rightarrow\operatorname{a.s.} 0$, where $\{Y_n\}$ is a solution of the homogeneous equation $y_n = a_1y_{n - 1} + a_2y_{n - 2} + \cdots + a_ky_{n - k}$. Several applications of possible theoretical interest are described. Also, the results suggest some extensions of classical results in the theory of random walks which are outlined.
Citation
James B. MacQueen. "A Linear Extension of the Martingale Convergence Theorem." Ann. Probab. 1 (2) 263 - 271, April, 1973. https://doi.org/10.1214/aop/1176996979
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