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<p<2, measure onR k for all 1≤k<∞. For functionsf, g such thatΛ p (|X(x−X(u)|) >g(u−s) andΛ p (|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(X 1+···+X n )),n≥1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX 1,X 2, ... are independent copies ofX andΛ p (Z)=sup t<0 t pP{|Z|<t} denotes the weakpth moment of a random variable Z.
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Bloznelis, M. Central limit theorem for stochastically continuous processes. Convergence to stable limit. J Theor Probab 9, 541–560 (1996). https://doi.org/10.1007/BF02214074
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DOI: https://doi.org/10.1007/BF02214074