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Title:First Exit Times Through Curvilinear Boundaries for Stochastic Sequences
Author(s):Crabtree, James Claude
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Let [Special characters omitted.] be a stochastic sequence and let [Special characters omitted.] be a predictable sequence. Define S(,n) for each n by S(,n) = X(,0) + X(,1)V(,1) + ... + V(,n)X(,n). Let [Special characters omitted.] be an increasing sequence of positive numbers and let [Special characters omitted.] Let 0 (LESSTHEQ) (alpha) (LESSTHEQ) 2 and let L be a positive Borel function. For each n, define W(,n) by W(,n) = (VBAR)V(,n)(VBAR)('(alpha))L(b(,n)/(VBAR)V(,n)(VBAR)) if V(,n) (NOT=) 0, W(,n) = 0 if V(,n) = 0. Under certain conditions on the b(,n)'s involving (alpha) and under certain conditions on the first two conditional truncated moments of the V(,n)X(,n)'s involving (alpha) and W(,n)'s we show there exist positive constants q(,0), c(,0), c(,1), and c(,2) such that [Special characters omitted.] for all (lamda) > 0. Furthermore, q(,0), c(,0), c(,1), and c(,2) do not depend on [Special characters omitted.] except through the conditions mentioned above.
When the (VBAR)V(,n)(VBAR)'s are nonrandom, the W(,n)'s are nonrandom so that our results give [Special characters omitted.] The conditions under which we prove these results are broad enough to include special cases for which [Special characters omitted.] is a sequence of i.i.d. random variables in the domain of attraction of a stable law and for which [Special characters omitted.] is a martingale difference sequence meeting the normed Marcinkiewicz-Zygmund conditions. The conditions on [Special characters omitted.] are rather general. For the lower bound on [Special characters omitted.] we require that [Special characters omitted.] For the upper bound, our conditions are met if W(,n)b(,n)('2-(alpha))/(b(,n)('2)-b(,n-1)('2)) is sufficiently large for each n.
We also use our results in the case b(,n) = b for each n (b some positive constant) to prove some a.s. convergence results.
Issue Date:1984
Description:124 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984.
Other Identifier(s):(UMI)AAI8502115
Date Available in IDEALS:2014-12-16
Date Deposited:1984

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