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Weak convergence of compound stochastic process, I

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  • Iglehart, Donald L.

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.

Suggested Citation

  • Iglehart, Donald L., 1973. "Weak convergence of compound stochastic process, I," Stochastic Processes and their Applications, Elsevier, vol. 1(1), pages 11-31, January.
  • Handle: RePEc:eee:spapps:v:1:y:1973:i:1:p:11-31
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    Cited by:

    1. Iksanov, Alexander, 2013. "Functional limit theorems for renewal shot noise processes with increasing response functions," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1987-2010.
    2. repec:eee:spapps:v:128:y:2018:i:2:p:505-544 is not listed on IDEAS
    3. Torrisi, Giovanni Luca, 2013. "Functional strong law of large numbers for loads in a planar network model," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 718-723.

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