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Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 2, Pages 360–366
(Mi tvp2236)
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This article is cited in 2 scientific papers (total in 2 papers)
Short Communications
On a minimax analogue of the weak law of large numbers
B. G. Pittel' Leningrad
Abstract:
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 $a=\operatornamewithlimits{val}_{u,v}\mathbf Mx^{u,v}$.
Received: 25.12.1969
Citation:
B. G. Pittel', “On a minimax analogue of the weak law of large numbers”, Teor. Veroyatnost. i Primenen., 16:2 (1971), 360–366; Theory Probab. Appl., 16:2 (1971), 361–367
Linking options:
https://www.mathnet.ru/eng/tvp2236 https://www.mathnet.ru/eng/tvp/v16/i2/p360
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