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By O. V. Gulinsky, A. Yu Veretennikov

In line with the Cramer-Chernoff theorem, which bargains with the "rough" logarithmic asymptotics of the distribution of sums of autonomous, identically random variables, this paintings essentially methods the extensions of this thought to established and, specifically, non-Markovian circumstances on functionality areas. Recurrent algorithms of identity and adaptive regulate shape the most examples in the back of the massive deviation difficulties during this quantity. the 1st a part of the ebook exploits a few rules and ideas of the martingale technique, specifically the concept that of the stochastic exponential. the second one half covers Freidlin's strategy, in line with the Frobenius-type theorems for confident operators, whuch end up to be powerful for the instances in attention. The ebook will be of price and curiosity to scientists within the box of chance, information and electric engineering, in addition to physicists facing statistical mechanics.

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Large deviations for discrete-time processes with averaging

In keeping with the Cramer-Chernoff theorem, which offers with the "rough" logarithmic asymptotics of the distribution of sums of autonomous, identically random variables, this paintings basically methods the extensions of this idea to based and, particularly, non-Markovian instances on functionality areas. Recurrent algorithms of id and adaptive keep an eye on shape the most examples at the back of the massive deviation difficulties during this quantity.

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D. random variables with a common distribu­ tion /x, Efi = 0, that are defined on a probability space (O, F, P) and that take values in R. ,£*, Fk = •••>£*)• Assume that the moment generating function (Cramer’s condi­ tion) Eexp (A£i) is finite for all A e R and the logarithmic moment generating function G^X) = In Eexp(A£i), satisfies the regularity condition lim |A|—»oo GjXA) |A| (R ) Consider a process X k - X k-i = 7(6(-^- i ) + Ik), A'o = a, * > 1. with b: R —>R and 7 = 1/n. Introduce a family of processes X n = (A'")o

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. Fi _1). d variables and aj(Xn'j~l) is Fj_i-measurable we have \n£k(nX,Xn'k~l) = £ A ^ A ^ " 1) + £ G ^ X ^ X ^ ) ) , j =i ;=i where In E (eAJ<,>(Jfn,J_l |Fj_i) = GM(Ai ffj (XBJ'"1)).

T the measure Qn'x. Nevertheless, we can represent f j <7n(Xfi s)) dW? t. Qn'x): * U n( X ^ 3)) d W : n M = n - 'Z < T k( Xn'k- 1) 6 *=1 M = n- 1 - E k=l +n ~ ' [j r £ n-X(

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