Limit theorems for renewal shot noise processes with eventually decreasing response functions

作者：Iksanov Alexander^*; Marynych Alexander; Meiners Matthias

来源：Stochastic Processes and Their Applications, 2014, 124(6): 2132-2170.

DOI：10.1016/j.spa.2014.02.007

• Finite-dimensional convergence
• Fractionally integrated stable Levy motion
• Fractionally integrated inverse stable subordinator
• Renewal shot noise process

We consider shot noise processes (X (t))(t %26gt;= 0) with deterministic response function h and the shots occurring at the renewal epochs 0 = S-0 %26lt; S-1 %26lt; S-2 ... of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X (ut))(u %26gt;= 0) as t -%26gt; infinity in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X (ut))(u %26gt;= 0) converge weakly as t -%26gt; infinity. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S-1 is in the domain of attraction of an alpha-stable law and the response function is regularly varying at infinity with index -beta (with 0 %26lt;= beta %26lt; 1/alpha or 0 %26lt;= beta %26lt;= alpha, depending on whether ES1 %26lt; infinity or ES1 = infinity), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut))(u %26gt;= 0). The limits are fractionally integrated stable Levy motions if ES1 %26lt; infinity and fractionally integrated inverse stable subordinators if ES1 = infinity.

• 出版日期2014-6