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Erschienen in:
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2016 | OriginalPaper | Buchkapitel

1. Perturbed Random Walks

verfasst von : Alexander Iksanov

Erschienen in: Renewal Theory for Perturbed Random Walks and Similar Processes

Verlag: Springer International Publishing

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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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Nächstes Kapitel Perpetuities
Fußnoten
1
We use x ∨ y or max(x, y), x ∧ y or min(x, y) interchangeably, depending on typographical convenience.
 
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
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
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
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
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
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
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
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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Metadaten
Titel
Perturbed Random Walks
verfasst von
Alexander Iksanov
Copyright-Jahr
2016
Buch
Renewal Theory for Perturbed Random Walks and Similar Processes

Print ISBN: 978-3-319-49111-0

Electronic ISBN: 978-3-319-49113-4

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-49113-4_1