Some bounds for the expected number of level crossings of certain harmonizable infinitely divisible processes.
Item
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Title
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Some bounds for the expected number of level crossings of certain harmonizable infinitely divisible processes.
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Identifier
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AAI9605661
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identifier
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9605661
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Creator
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Shen, Kevin.
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Contributor
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Adviser: Michael B. Marcus
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Date
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1995
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Let Y = {dollar}\{lcub}Y(t),t \in \ \lbrack 0,1\rbrack\{rcub}{dollar} be a harmonizable, symmetric, infinitely divisible stochastic process, and let {dollar}N\sb{lcub}u{rcub}\lbrack 0,1\rbrack{dollar} be the number of crossings at level u by Y during the time interval (0,1). The expected value of {dollar}N\sb{lcub}0{rcub}\lbrack 0,1\rbrack{dollar} and the asymptotic behavior of {dollar}EN\sb{lcub}u{rcub}\lbrack 0,1\rbrack,{dollar} as {dollar}u \to \infty{dollar}, are studied in this dissertation.;Let {dollar}\xi{dollar} be a symmetric real valued infinitely divisible random variable with characteristic function{dollar}{dollar}Ee\sp{lcub}iu\xi{rcub} = e\sp{lcub}-\psi{rcub}(\vert u\vert)\cr{dollar}{dollar}.;where {dollar}\psi(u){dollar} = {dollar}\int\sbsp{lcub}0{rcub}{lcub}\infty{rcub}{dollar}(cos {dollar}ut - \ 1)d\tau\lbrack t,\infty), \tau{dollar} is a Levy measure defined on {dollar}R\sp+{dollar}, i.e., {dollar}\vert \int\sbsp{lcub}0{rcub}{lcub}\infty{rcub}(1 \wedge t\sp2)d\tau\lbrack t,\infty)\vert < \infty{dollar} and {dollar}\psi\sb{lcub}g{rcub}(\vert u\vert){dollar} = {dollar}E\sb{lcub}g{rcub}\{lcub}\psi(\vert ug\vert)\{rcub}{dollar}, where g is a standard normal random variable. Let Y be the real part of the process {dollar}\{lcub}X(t),t \in \ \lbrack 0,1\rbrack \{rcub}{dollar}, which is defined by{dollar}{dollar}E\exp iRe\left(\sum\sbsp{lcub}j=1{rcub}{lcub}n{rcub} \bar\alpha\sb{lcub}j{rcub}X(t\sb{lcub}j{rcub})\right) = \exp - \int \psi\sb{lcub}g{rcub}(\vert\sum\sbsp{lcub}j=1{rcub}{lcub}n{rcub}\bar \alpha\sb{lcub}j{rcub}e\sp{lcub}it\sb{lcub}j{rcub}\sp{lcub}\lambda{rcub}{rcub}\vert)dF(\lambda){dollar}{dollar}.;where {dollar}\alpha\sb{lcub}1{rcub},\cdots,\alpha\sb{lcub}n{rcub}{dollar} are finite, {dollar}t\sb{lcub}1{rcub},\cdots,t\sb{lcub}n{rcub} \in{dollar} (0,1), for all integer {dollar}n >{dollar} 0, and {dollar}\lambda{dollar} is a positive random variable with distribution function F.;Equivalently, the process Y has an alternate representation as a stochastic integral:{dollar}{dollar}Y(t) = \int\sbsp{lcub}0{rcub}{lcub}\infty{rcub}\cos\lambda t\ dM(F(\lambda)) + \int\sbsp{lcub}0{rcub}{lcub}\infty{rcub}\sin\lambda t\ dM\sp\prime(F(\lambda)){dollar}{dollar}.;where M and {dollar}M\sp{lcub}\prime{rcub}{dollar} are independently scattered infinitely divisible measures determined by {dollar}\psi\sb{lcub}g{rcub}{dollar}.;Under regularity conditions on {dollar}\psi{dollar} or equivalently on {dollar}\tau,{dollar} the following results are obtained:;(1) There exist constants 0 {dollar}< c\sb{lcub}1{rcub},c\sb{lcub}2{rcub} < \ \infty,{dollar} such that.;{dollar}{dollar}c\sb{lcub}1{rcub}\Vert\lambda\Vert\sb{lcub}\psi{rcub} \le EN\sb{lcub}0{rcub}\lbrack 0,1\rbrack \le c\sb{lcub}2{rcub}\Vert\lambda\Vert\sb{lcub}\psi{rcub}{dollar}{dollar}.;where {dollar}\Vert\lambda\Vert\sb{lcub}\psi{rcub} \sbsp{lcub}={rcub}{lcub}\rm def{rcub}{dollar} inf {dollar}\{lcub} c > \ 0 : E\psi({lcub}\lambda\over c{rcub}) \le 1\{rcub}.{dollar}.;(2) When {dollar}EN\sb{lcub}0{rcub}\lbrack 0,1\rbrack < \infty{dollar},;{dollar}{dollar}\lim\limits\sb{lcub}u\to\infty{rcub} {lcub}EN\sb{lcub}u{rcub}\lbrack 0,1\rbrack\over\tau\lbrack u,\infty){rcub} = {lcub}\sqrt{lcub}2\sp{lcub}p{rcub}{rcub}\Gamma({lcub}p\over2{rcub} + 1\over\pi{rcub} \int\sbsp{lcub}0{rcub}{lcub}\infty{rcub}\lambda dF(\lambda),{dollar}{dollar}.;where {dollar}\Gamma{dollar} is the gamma function.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
