具有增响应函数的更新散粒噪声过程的泛函极限定理
Functional limit theorems for renewal shot noise processes with increasing response functions
摘要：
我们考虑更新散粒噪声过程的响应函数最终是非减的，并定期在无穷大变化。本文证明了更新散粒噪声过程在$J_1$或$M_1$拓扑下，在空间$D[0,\infty)$中适当地归一化和中心化后的弱收敛性.极限过程要么是谱非正稳定的Lévy过程，包括布朗运动，要么是逆稳定从属（当响应函数缓慢变化时），要么是分数积分稳定过程或分数积分逆稳定从属（当正则变差指数为正时）。证明利用了更新过程的优良性质、稳定Lévy过程的分布性质和连续映射定理。
英文摘要：
We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space $D[0,\infty)$ under the $J_1$ or $M_1$ topology. The limiting processes are either spectrally nonpositive stable Lévy 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 Lévy processes and the continuous mapping theorem.
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