Functional limit theorems for renewal shot noise processes

Details Functional limit theorems for renewal shot noise processes Functional limit theorems for renewal shot noise processes

2012-02-09

arxiv.org/abs/1202.1950v1

Mathematics

Probability

Scientific paper

We consider renewal shot noise processes with response functions which are regularly varying at infinity with nonnegative index. We prove weak convergence of, properly normalized and centered, renewal shot noise processes in the space $D[0,\infty)$ under the $J_1$ or $M_1$ topology. The limiting processes are either spectrally nonpositive stable L\'{e}vy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of regular variation is positive). The proof exploits fine properties of renewal processes, distributional properties of stable L\'{e}vy processes and the continuous mapping theorem.

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