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Strong law of large numbers and mixing for the invariant distributions of measure-valued diffusions

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  • Pinsky, Ross G.

Abstract

Let denote the space of locally finite measures on Rd and let denote the space of probability measures on . Define the mean measure [pi][nu] of byFor such a measure [nu] with locally finite mean measure [pi][nu], let f be a nonnegative, locally bounded test function satisfying =[infinity]. [nu] is said to satisfy the strong law of large numbers with respect to f if / converges almost surely to 1 with respect to [nu] as n-->[infinity], for any increasing sequence {fn} of compactly supported functions which converges to f. [nu] is said to be mixing with respect to two sequences of sets {An} and {Bn} ifconverges to 0 as n-->[infinity] for every pair of functions f,g[set membership, variant]Cb1([0,[infinity])). It is known that certain classes of measure-valued diffusion processes possess a family of invariant distributions. These distributions belong to and have locally finite mean measures. We prove the strong law of large numbers and mixing for many such distributions.

Suggested Citation

  • Pinsky, Ross G., 2003. "Strong law of large numbers and mixing for the invariant distributions of measure-valued diffusions," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 117-137, May.
    • Handle: RePEc:eee:spapps:v:105:y:2003:i:1:p:117-137
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    References listed on IDEAS

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    1. Dawson, Donald A. & Fleischmann, Klaus, 1997. "Longtime behavior of a branching process controlled by branching catalysts," Stochastic Processes and their Applications, Elsevier, vol. 71(2), pages 241-257, November.
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