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

Functional limit theorems for renewal shot noise processes with increasing response functions