On a~minimax analogue of the weak law of large numbers

B. G. Pittel'

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 360-366

Résumé

Let $U$ and $V$ be two finite sets and, for any $u_1,u_2,\dots\in U$, $v_1,v_2,\dots\in V$, $x_t^{u_t,v_t}$ be independent non-negative random variables with distribution functions $F_{u_t,v_t}(x)$, $t=1,2,\dots$ respectively. At each time $t=1,\dots,n$ the first player chooses a probability distribution of $u_t$ depending on the observed data $x_1^{u_1,v_1},\dots,x_{t-1}^{u_{t-1},v_{t-1}}$. The second player makes his “move”: chooses a distribution for $v_t$ in the same way. Put $$ w_n(x)=\sup\inf\mathbf P\{x_1^{u_1,v_1}+\dots+x_n^{u_n,v_n}\le nx\} $$ where supremum is taken over all the strategies of the first player and infimum over all the strategies of the second player. The main result of the paper (Theorem 1) is: For any $\varepsilon>0$, $w_n(a+\varepsilon)\to1$, $w_n(a-\varepsilon)\to0$ where $a=\operatornamewithlimits{val}_{u,v}\mathbf Mx^{u,v}$.