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Renewal Theory for Perturbed Random Walks and Similar Processes pp 43–86Cite as

Birkhäuser

Perpetuities

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Abstract

Let \((M_{k},Q_{k})_{k\in \mathbb{N}}\) be independent copies of an \(\mathbb{R}^{2}\)-valued random vector (M, Q) with arbitrary dependence of the components, and let X 0 be a random variable which is independent of \((M_{k},Q_{k})_{k\in \mathbb{N}}\).

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Notes

  1. 1.

    Among other things this implies \(\mathbb{E}\log ^{+}\vert Q\vert = \infty\).

  2. 2.

    Condition (A6) together with the second part of (A1) ensures that \(\#\{k:\theta _{ k}^{(n)} \leq T\} \geq 1\).

  3. 3.

    Although a j, n ’s depend on t we suppress this dependence for the sake of clarity.

  4. 4.

    Condition (2.54) is only used in this part of the proof.

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    Alexander Iksanov

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Iksanov, A. (2016). Perpetuities. In: Renewal Theory for Perturbed Random Walks and Similar Processes. Probability and Its Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49113-4_2

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