On Sample Functions Behavior of Stable Processes
نویسنده
چکیده
We investigate the asymptotic behavior of sample functions of stable processes when t→∞. We compare our results with the iterated logarithm law, results for the first hitting time and most visited sites problems. 1 Main Notions 1. Let X1, X2... be mutually independent random variables with the same law of distribution F (x). The distribution F (x) is called strictly stable if the random variable X = (X1 +X2 + ...+Xn)/n 1/α (1.1) is also distributed according to the law F (x). The number α (0 < α≤2) is called a characteristic exponent of the distribution. When α 6=1 the characteristic function of X has the form (see[2]) E[exp (iξX)] = exp {−λ|ξ|[1− iβ(signξ)(tan πα 2 )]}, (1.2) where −1≤β≤1, λ > 0. When α = 1 we have E[exp (iξX)] = exp {−λ|ξ|[1 + 2iβ π (signξ)(log |ξ|)]}, (1.3) where −1≤β≤1, λ > 0. The homogeneous process X(τ)(X(0) = 0) with independent increments is called a stable process if E[exp (iξX(τ))] = exp {−τ |ξ|[1− iβ(signξ)(tan πα 2 )]}, (1.4) where 0 < α < 2, α 6=1,−1≤β≤1, τ > 0. When α = 1 we have E[exp (iξX(τ))] = exp {−τ |ξ|[1 + 2iβ π (signξ)(log |ξ|)]}, (1.5)
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تاریخ انتشار 2006