Filtered Brownian motions as weak limit of filtered Poisson processes (Q1781188)

Let \(N^\lambda_s\) be a Poisson process of constant intensity \(\lambda\), \(\widehat N^\lambda_s= \lambda^{-1/2}(N^\lambda_s-\lambda s)\), and let \(K(t, s)\) be a kernel satisfying certain regularity conditions. It is proved that the sequence of processes \(Y^n_t= \int^t_0K(t,s)\,d\widehat N^n_s\) converges weakly in a Hölderian space to the process \(Y_t= \int^t_0 K(t, s)\,dB_t\), where \(B_t\) is the Brownian motion. In particular, when \(K(t, s) = (t-s)^{H-1/2}\), \(H\in(0, 1)\), one gets the weak convergence of the corresponding shot noise process to the fractional Brownian motion. The proof is based on an application of a convergence of Hilbert space valued semimartingales and a use of a radonification result of \textit{A. Jakubowski}, \textit{S. Kwapień}, \textit{P. Raynaud de Fitte} and \textit{J. Rosiński} [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, No. 3, 429--440 (2002; Zbl 1056.60001)].