Abstract
In this chapter, we collect some overview sections of topics which are closely related to the fixed-point equation.
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Notes
- 1.
See p. 47 for the definition.
- 2.
Benda [36] (see also Peigné and Woess [226]) proved that, if a general Markov chain is recurrent and locally contractive, then it has an invariant measure which is unique up to a multiplicative constant.
- 3.
The proof in [22] contains a gap, which was closed by Brofferio [63] and independently, in a more general setting, by Benda [36].
- 4.
Notice that we have \(\mathbb {E}[\log A]=0\). In the literature such an equation is called Poisson equation. This is in contrast to equation (2.4.61) which we called a renewal equation where the random variable has a nonzero drift.
- 5.
The existence of the limit follows by an application of the subadditive ergodic theorem; see Kingman [178].
- 6.
See p. 260 for the definition of a subexponential distribution.
- 7.
A random variable is non-lattice if is not supported on any of the sets \(a{\mathbb {Z}}+b\), \(a>0\). Observe that this definition is stronger than non-arithmeticity.
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© 2016 Springer International Publishing Switzerland
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Buraczewski, D., Damek, E., Mikosch, T. (2016). Miscellanea. In: Stochastic Models with Power-Law Tails. Springer Series in Operations Research and Financial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-29679-1_5
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DOI: https://doi.org/10.1007/978-3-319-29679-1_5
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