By O. V. Gulinsky, A. Yu Veretennikov

In response to the Cramer-Chernoff theorem, which bargains with the "rough" logarithmic asymptotics of the distribution of sums of self reliant, identically random variables, this paintings basically techniques the extensions of this concept to based and, specifically, non-Markovian circumstances on functionality areas. Recurrent algorithms of id and adaptive keep an eye on shape the most examples at the back of the big deviation difficulties during this quantity. the 1st a part of the publication exploits a few rules and ideas of the martingale procedure, particularly the concept that of the stochastic exponential. the second one half covers Freidlin's method, in keeping with the Frobenius-type theorems for optimistic operators, whuch turn out to be potent for the instances in attention. The publication can be of worth and curiosity to scientists within the box of likelihood, records and electric engineering, in addition to physicists facing statistical mechanics.

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With m - s [Az - G„(A)]. 1. Assume that (f*) are Gaussian with E ip is absolutely continuous, y>o Xy otherwise. 2. The result remains valid if condition (i) is replaced by condition of linear growth. O. Gulinsky and A. Veretennikov 54 Stochastic exponent and cummulant. Transformation of measure There are several ways in which one can go about studying large deviations for the sequence X n. We choose one which emphasizes the role of the stochastic moment generating function and the stochastic exponential.

Thus, by letting L -> oo, we get the required result. Pukhalskii (1991) proved for large deviations an analog of Prokhorov theorem on the equivalence of weak relative compactness and tightness for a family of probability measures. In our presentation in the rest of this section we follow Pukhalskii (1991) and Liptser and Pukhalskii (1992). We need the following definition. 6. The sequence (Fn), n > 1, is said to have the partial large deviation principle if any subsequence {n') of (n) contains a further subsequence (n") such that {Fnn) obeys the large deviation principle with some rate function.

Let Cramer’s condition, the regularity condition (R) and con­ ditions (i)-(iii) hold. Then the sequence X n, n > 1, obeys the large deviation principle in D[0,1] with the rate function - % [ 0,a])D d£, (p is absolutely continuous, (p0 = x, /(y,) = J [ oo, otherwise . with m - s [Az - G„(A)]. 1. Assume that (f*) are Gaussian with E ip is absolutely continuous, y>o Xy otherwise. 2. The result remains valid if condition (i) is replaced by condition of linear growth.

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