By Linkov YU. N.
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Extra info for Asymptotic statistical methods for stochastic processes
Lim it th eorem s for sem im artingales C entral lim it th eorem for loca l m artingales. 1) y* = M* + Cf* * (#** - i/*) , where M l G AIfoc 0(Ft, P t), /z* is an F*-optional integral-valued random measure on R + x Ro for which the measure satisfies the property of P (F* )-
1. Consider a fc-dimensional local martingale Y* = (Y *’1, . . 9) where M* = (M*’1, . 1), u* = (u *'1, . . , V * y € aioc ( s , p* ), and u 1* - «/*) = ( u ‘ -1 * (/i* - I/*) , . . , Ul'k * ( a** - */*))'. 2. Suppose the following conditions be fulfilled for a fixed s E (0, oo) and any e E (0,1]: 1) PM imt_*oox(|t/t|>e)|t/t|*I4 = 0; 2) P t-lim (_ 00[(M t,M t) st+x(|(7t|< e)£P(£/t)/ *l/it] = H s)> where I(s) is some symmetric nonrandom k x k-matrix dependent on s mth finite elements. 10) C (I^ IP 4) ^ Af( 0, /( « ) ) , t -> oo.
Prom this, in view of . m + zin) (A - l) * (n - v) has a unique up to P-indistinguishability solution € Ad(F, P) with E z£ ) = 1 in the class of nonnegative local martingales A4j£c(F, P). We introduce probability measures P K with dPK = z^ d P . 2. 7) it follows that the measures P In view of are such that p ^ ’° = p ° and £Tn G