Central limit theorem for stochastically continuous processes. Convergence to stable limit

Abstract

LetX={X(t), t∈[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0 g(u−s) andΛp(|X(s−X(t|)∧|X(t)−X(u|)>f(u−s), 0 ≤s ≤t ≤u ≤ 1, conditions are found which imply that the distributions −(n−1/p(X1+···+Xn )),n≥1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX1,X2, ... are independent copies ofX andΛp(Z)=supt<0tpP{|Z|<t} denotes the weakpth moment of a random variable Z.