On a game random walk on the line

B. G. Pittel'

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 632-643

Résumé

Let $I$ and $J$ be two finite sets, and $X_{ij}(r)$, $Y_{ij}(r)$ ($r=1,2,\dots,(i,j)\in I\times J$) be random variables (independent for different values of $r$) with distribution functions $G_{ij}(x)$ and $F_{ij}(x)$ respectively. The game consists of a sequence of sets. At each set $r=1,2,\dots,$ player 1 (player 2) chooses a probability distribution on $I(J)$ depending on observed values of $X_0,Z_0,\dots,X_{r-1},Z_{r-1}$, where $$ X_\alpha=X_{\alpha-1}+X_{i_\alpha j_\alpha},\quad Z_\alpha=Z_{\alpha-1}-Y_{i_\alpha j_\alpha},\quad\alpha=1,2,\dots;\quad X_0=0,\quad Z_0=t $$ ($t$ is the initial resource of the game), $\nu$ is the stochastic duration of the game equal to $\min\{s,\min(r\mid Z_r<0)\}$ ($s$ is the maximal duration of the game given a priori) and the total gain of player 1 is $X_\nu$. Existence of the value of the game $H_s(t)$ is proved. Under general assumptions, it is proved that $\widehat H(t)=\lim\limits_{s\to\infty}H_s(t)$ is a special solution of a minimax analogue of a renewal equation and $$ \widehat H(t)=\lambda t+o(t),\quad\operatorname{val}(EX_{ij}-\lambda EY_{ij})=0,\quad\lambda>0. $$ It is also proved that $\widehat H(t)$ is the unique solution of this equation in the class of all quasilinear (non-negative quasilinear) solutions provided $\min\limits_{i,j}EY_{ij}>0(\sup\limits_{\gamma>-\lambda}\min\limits_{i,j}(EX_{ij}+\gamma EY_{ij})>0)$, and $\widehat H(t)=\lambda t+o(t^\varepsilon)$ for any $\varepsilon>0$, if, in addition $$ E(Y_{ij}^+)^2<\infty,\quad Y_{ij}^+=\max(0,Y_{ij}),\quad(i,j)\in I\times J. $$