University of Illinois Urbana-Champaign

First Exit Times Through Curvilinear Boundaries for Stochastic Sequences

Crabtree, James Claude

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https://hdl.handle.net/2142/71223

Description

Title
First Exit Times Through Curvilinear Boundaries for Stochastic Sequences
Author(s)
Crabtree, James Claude
Issue Date
1984
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
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.
Graduation Semester
1984
Type of Resource
text
Permalink
http://hdl.handle.net/2142/71223

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