By V. S. Vladimirov
Vladimirov V.S. Equations of mathematical physics (M.Dekker, 1971)(ISBN 0824717139)(1s)_MCde_
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Extra info for Equations of mathematical physics
Example text
Xj^} in the sequence Zi(co), . . , X„(co) (with cyclic counting). The contribution of the cyclic terms F^"^(co), J5G{« — A: + 2 , . . , « } , to jR„(co,E) is o(n) as « -^ oo. i^„(ft), •) is a natural generalization of the empirical measure L„{co, •) and of the empirical pair measure M„(af, •). 34) R„(CD, I ) = L„(CD, {x,}) = L„,,(a;). }) = M„jj(co). Thus M„(co, ') is the two-dimensional marginal of i^„(co, •). s. 8]. Let Z^, . . 36) ^3 = {PeJ^M'^rnaxJP{^,} - />,{ZJ| > e}, Let Qi^^ be the P^-distribution of i^„ on ^^(Q).
Introduction to Large Deviations . . , « — 1} and to be the cyclic ordered pair {X^(co), X^ (co)) if P = n. For each subset {x^,Xy} of F^, let M„JJ((JO) be \/n times the number of pairs {Y^p''\co)} for which ^^"^(co) = (x^,x,)'; thus ^G{1,2, 1 " For each oj, the numbers {M„^ j(co)} define a probabiHty measure M„(co, •) on the set of all subsets of F^. ){^}. The measure M„((D, •) is called the empirical pair measure corresponding to . . ,X^{oj). 23) r L^,i{cD) = Y, M„jj((o) = X M„^k,i(^) for each / = 1, .
X„(co)) periodically into a doubly infinite sequence, obtaining a point X(n,co) = (... X,(ojlX,iojlX,(col.. ) in Q; (X(n,co))^ = X^(co), (X(n,cjo))2 = ^li^)^ probability measure on ^(Q) by (1-33) ^tc. For each COGQ, define a ^>,-) = ^"f Vm«)(-), where T^ is the identity mapping and T^ = T(T^~^) for /c = 2 , . . , « - 1. 6. Level-3: Empirical Process 23 each Borel subset B of Q, R„(a),B) is the relative frequency with which X(n,(jo), TX{n,co), . . , r""^X(w, co) is in B. oS) is periodic of period n, R(n, •) is for each co a strictly stationary probabiHty measure.