ИСТИНА

We consider random walk $S_n=\sum_{i=1}^{n} X_i$ with i.i.d. steps $X_i$, satisfying right-hand Cramer condition and $MX \geq 0$. Komlos, Tusnady (1975) have proved the conditional large deviation theorem for $S_{n,m}\geq \theta n$, $S_{n,m}:=S_{n+m}-S_{m}$, under the condition $S_{n,i}<\theta n, i\leq m$, $m,n\rightarrow\infty$. This result can be looked at as a generalization of Petrov's Theorem for large deviations $S_n\geq \theta n$. Our communication concerns conditional large deviation theorem for $M_{n,m}:=\max_{i\leq n} S_{i,m}\geq \theta n$ under the condition $M_{n,i}< \theta n, i\leq m$. Starting with a problem of finding asymptotics of probabilities $P(M_{n,0}\geq \theta n)$, we pass to probabilities $P(M_{n,n}\geq \theta n| M_{n,i}<\theta n, i\leq n)$. Further it is shown that the last probability is equivalent to the probability in question. Finally, we get for it the following asymptotics: $$C(\theta) e^{-\Lambda(\theta)n} n^{-1/2},\ m,n\rightarrow\infty. $$ The expression for multiplier $C(\theta)$ is given in form of rather sofisticated functional of random walk trajectory. This result is a base for limit theorem (including large deviations) for Shepp statistics $\max_{i\leq n} \max_{j\leq m} S_{i,j}.$