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I consider an integral (or a sum with random index) $$ M(t) =\int\limits_0^t f(t-u)dX(u), $$ where $$ X(u) = \sum\limits_{i=1}^{N(u)} \xi_i,\qquad N(u)=\max\{k: \tau_1+\,\dots,\,\tau_k\, <\, u\}, $$ $$ \{\tau_i,\xi_i\}_{i=1}^\infty\quad \mbox{i.i.d.r. elements} $$ $$ \tau_i\mbox{ and } \xi_i \mbox{ are depending on each other}. $$ Function $f(u)$ has a heavy tail. $$ f(u)\sim u^{-\alpha}, \quad \alpha\in{(0,1)} $$ What is known about asymptotic of
$$ \mathsf{Var} M(t)\mbox{ as }t\to\infty? $$ Are anywhere Limit Theorems for $M(t)$?

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Sorry, I don't have the answer. But, this belongs to the generalized shot noise studied by John Rice. This might help:

  • John Rice. On Generalized Shot Noise. Advances in Applied Probability. Vol. 9, No. 3 (Sep., 1977) , pp. 553-565 [jstor.org]
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