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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1 Answer
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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]
