Weak convergence of compound stochastic process, I

Donald L. Iglehart

Stochastic Processes and their Applications, 1973, vol. 1, issue 1, 11-31

Abstract: Compound stochastic processes are constructed by taking the superpositive of independent copies of secondary processes, each of which is initiated at an epoch of a renewal process called the primary process. Suppose there are M possible k-dimensional secondary processes {[xi]v(t):t[greater-or-equal, slanted]0}, v=1,2,...,M. At each epoch of the renewal process {A(t):t[greater-or-equal, slanted]0} we initiate a random number of each of the M types. Let ml:l[greater-or-equal, slanted]1} be a sequence of M-dimensional random vectors whose components specify the number of secondary processes of each type initiated at the various epochs. The compound process we study is (t)=[summation operator]l=1A(t)[summation operator]v=1M[summation operator]j=1Mlv[xi]ljv(t-Tl), t[greater-or-equal, slanted]0, where the [xi]vlj([radical sign]) are independent copies of [xi]v,mlv is the vth component of m and {[tau]l:l[greater-or-equal, slanted]1} are the epochs of the renewal process. Our interest in this paper is to obtain functional central limit theorems for {Y(t):t[greater-or-equal, slanted]0} after appropriately scaling the time parameter and state space. A variety of applications are discussed.

Keywords: Compound; stochastic; processes; functional; central; limit; theorem; invariance; principle; weak; convergence (search for similar items in EconPapers)
Date: 1973
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Citations: View citations in EconPapers (7)

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