By O. V. Gulinsky, A. Yu Veretennikov
In accordance with the Cramer-Chernoff theorem, which offers with the "rough" logarithmic asymptotics of the distribution of sums of autonomous, identically random variables, this paintings essentially ways the extensions of this thought to based and, particularly, non-Markovian circumstances on functionality areas. Recurrent algorithms of identity and adaptive keep an eye on shape the most examples at the back of the massive deviation difficulties during this quantity. the 1st a part of the booklet exploits a few rules and ideas of the martingale procedure, specially the idea that of the stochastic exponential. the second one half covers Freidlin's technique, in accordance with the Frobenius-type theorems for optimistic operators, whuch end up to be potent for the circumstances in attention. The ebook can be of worth and curiosity to scientists within the box of chance, information and electric engineering, in addition to physicists facing statistical mechanics.
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Extra info for Large deviations for discrete-time processes with averaging
Example text
5 |0 n —»oo We show that (0) and (II) or (II7) imply lim sup lim sup n _1 In Fn(y: p(x,y) < 8) < -I(x). $ |0 n —*oo To this end, pick 7 > 0 and note that the compact set $ (/(z ) - 7 ) does not contain x. Choose 6 > 0 such that 8 < 1/2p ($(I(x) - 7 ), x). Then, by condition (II') limsup limsup n ^ ln F J y : p(x,y) < 6) 6 |0 n —»oo < limsup limsup n~l In Fn(y: p($(I(x) - 7 ),y) > 8 ) < I(x) —27 . 5 |0 n—*oo Since 7 is arbitrary by letting 7 —>0 we obtain the desired result. 35) as the local large deviation principle.
We will prove (I). (II) is proved similarly. 32 O. Gulinsky and A. Veretennikov Let G c D be an open set. 36) and from (n') one can choose a subsequence (n,;) such that X n" obeys the large deviation principle with rate function /". Particularly, this means that lim inf 1 Tl" n" In P (A'"" 6 G) > - inf I"(v ). 37) However, we may choose (n") as the sequence (n") above. 37) lim inf n " 1 In P (A" 6 G ) > - inf J( 5 |0 n—*oo Since 7 is arbitrary by letting 7 —>0 we obtain the desired result. 35) as the local large deviation principle. It would be convenient to reduce the analysis of large deviations to the local large deviations principle. However, generally speaking the local large deviation principle does not imply the full one. Example (Wentzell and Freidlin (1984)). Let X = R, p(x,y) = \x - y\ and Fn(x) = cN(Q,n~l) + cA/"(0, a(n)) with a(n) = exp(en2). 0 n —»oo / for every x e R. The function I(x) = x2/2, evidently, satisfies the condition (0).