A zero-sum two person game is repeatedly played. Some of the payoffs are "absorbing" in the sense that, once any of them is reached, all future payoffs remain unchanged. Let $v_n$ denote the value of the $n$-times repeated game, and let $v_\infty$ denote the value of the infinitely-repeated game. It is shown that $\lim v_n$ always exists. When the information structure is symmetric, $v_\infty$ also exists and $v_\infty = \lim v_n$.
"Repeated Games with Absorbing States." Ann. Statist. 2 (4) 724 - 738, July, 1974. https://doi.org/10.1214/aos/1176342760