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Renewal Theory for Perturbed Random Walks and Similar Processes pp 1–41Cite as

Birkhäuser

Perturbed Random Walks

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Part of the book series: Probability and Its Applications ((PA))

Abstract

Let \((\xi _{k},\eta _{k})_{k\in \mathbb{N}}\) be a sequence of i.i.d. two-dimensional random vectors with generic copy (ξ, η). No condition is imposed on the dependence structure between ξ and η.

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Notes

  1. 1.

    We use x ∨ y or max(x, y), x ∧ y or min(x, y) interchangeably, depending on typographical convenience.

  2. 2.

    To give a better feeling of the result, consider the simplest situation when \(\mathbb{E}\xi \in (-\infty, 0)\) and \(\mathbb{E}\eta ^{+} <\infty\). Then, by the strong law of large numbers, S n drifts to −∞ at a linear rate. On the other hand, lim n → ∞ n −1 η n + = 0 a.s. by the Borel–Cantelli lemma which shows that η n + grows at most sublinearly. Combining pieces together shows lim n → ∞ (S n−1 +η n ) = −∞ a.s.

  3. 3.

    A strange assumption \(\mathbb{P}\{\eta = -\infty \}\in [0, 1)\) which is made here and in Lemma 1.3.12 is of principal importance for the proof of Theorem 2.1.5.

  4. 4.

    Actually, Breiman (Proposition 3 in [52]) only proved the result for α ∈ (0, 1). The whole range α > 0 was later covered by Corollary 3.6 (iii) in [70].

  5. 5.

    Actually, \(\mathbb{E}\exp (a\,\sup _{n\geq 0}S_{n}) <\infty\) if, and only if, \(\mathbb{E}e^{a\xi } <1\). To prove the implication ⇐ just use the inequality \(\mathbb{E}\exp (a\,\sup _{n\geq 0}S_{n}) \leq \mathbb{E}\sum _{n\geq 0}e^{aS_{n}} = (1 - \mathbb{E}e^{a\xi })^{-1}\).

  6. 6.

    This is indeed a probability measure because, in view of the first condition in (1.16), \((e^{aS_{n}})_{n\in \mathbb{N}_{ 0}}\) is a nonnegative martingale with respect to the natural filtration.

  7. 7.

    The only principal difference is that one should use \(S_{[n\cdot ]}/n \Rightarrow \mu \Upsilon (t)\) on D where \(\Upsilon (t) = t\) for t ≥ 0, rather than Donsker’s theorem in the form (1.54).

  8. 8.

    We recall that \(\sup _{\lambda _{ n}(\theta _{k}^{(n)})\leq t}(\,f_{0}(\theta _{k}^{(n)}) + y_{k}^{(n)}) = f_{0}(0)\) and \(\sup _{\lambda _{ n}(\bar{\theta }_{i}^{(n)})\leq t}(\,f_{0}(\bar{\theta }_{i}^{(n)}) +\bar{ y}_{i}^{(n)}) = f_{0}(0)\) if the supremum is taken over the empty set.

  9. 9.

    The weak convergence of finite-dimensional distributions is immediate from K [nt]  ≤  + K [nt]  >  = [nt] and the fact that K [nt]  >  converges in distribution. This extends to the functional convergence because the limit is continuous and K [nt]  ≤  is a.s. nondecreasing in t (recall Pólya’s extension of Dini’s theorem: convergence of monotone functions to a continuous limit is locally uniform).

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Authors and Affiliations

  1. Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Alexander Iksanov

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Iksanov, A. (2016). Perturbed Random Walks. 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_1

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