arXiv subjects

Fabian Buckmann

Publications and source records attributed to Fabian Buckmann.

3 recordsLinked to original sources

arXiv2018

An Arcsine Law for Markov Random Walks

The classic arcsine law for the number $N_{n}^{>}:=n^{-1}\sum_{k=1}^{n}\mathbf{1}_{\{S_{k}>0\}}$ of positive terms, as $n\to\infty$, in an ordinary random walk $(S_{n})_{n\ge 0}$ is extended to the case when this random walk is governed by a positive recurrent Markov chain $(M_{n})_{n\ge 0}$ on a countable state space $\mathcal{S}$, that is, for a Markov random walk $(M_{n},S_{n})_{n\ge 0}$ with positive recurrent discrete driving chain. More precisely, it is shown that $n^{-1}N_{n}^{>}$ converges in distribution to a generalized arcsine law with parameter $ρ\in [0,1]$ (the classic arcsine law if $ρ=1/2$) iff the Spitzer condition $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\mathbb{P}_{i}(S_{n}>0)\ =\ ρ$$ holds true for some and then all $i\in\mathcal{S}$, where $\mathbb{P}_{i}:=\mathbb{P}(\cdot|M_{0}=i)$ for $i\in\mathcal{S}$. It is also proved, under an extra assumption on the driving chain if $0<ρ<1$, that this condition is equivalent to the stronger variant $$ \lim_{n\to\infty}\mathbb{P}_{i}(S_{n}>0)\ =\ ρ. $$ For an ordinary random walk, this was shown by Doney for $0<ρ<1$ and by Bertoin and Doney for $ρ\in\{0,1\}$.

math.PR↗

arXiv2017

Fluctuation theory for Markov random walks

Two fundamental theorems by Spitzer/Erickson and Kesten/Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk $(S_{n})_{n\ge 0}$ with iid increments $X_{1},X_{2},\ldots$ and the existence of moments of various related quantities like the first passage into $[x,\infty)$ and the last exit time from $(-\infty,x]$ for arbitrary $x\in\mathbb{R}_{\geqslant}$ are studied in the Markov-modulated situation when the $X_{n}$ are governed by a positive recurrent Markov chain $M=(M_{n})_{n\ge 0}$ on a countable state space $\mathcal{S}$, thus for a Markov random walk $(M_{n},S_{n})_{n\ge 0}$. Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks $(S_{τ_{n}(i)})_{n\ge 0}$, where $τ_{1}(i),τ_{2}(i),\ldots$ denote the successive return times of $M$ to state $i$, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the afore-mentioned theorems are not one-to-one extensions of those in the iid case and cannot be as illustrated by a number of counterexamples. In fact, various excursion measures will have to be introduced so as to characterize the existence of moments of different quantities.

math.PR↗

arXiv2017

Stability of perpetuities in Markovian environment

The stability of iterations of affine linear maps $Ψ_{n}(x)=A_{n}x+B_{n}$, $n=1,2,\ldots$, is studied in the presence of a Markovian environment, more precisely, for the situation when $(A_{n},B_{n})_{n\ge 1}$ is modulated by an ergodic Markov chain $(M_{n})_{n\ge 0}$ with countable state space $\mathcal{S}$ and stationary distribution $π$. We provide necessary and sufficient conditions for the a.s. and the distributional convergence of the backward iterations $Ψ_{1}\circ\ldots\circΨ_{n}(Z_{0})$ and also describe all possible limit laws as solutions to a certain Markovian stochastic fixed-point equation. As a consequence of the random environment, these limit laws are stochastic kernels from $\mathcal{S}$ to $\mathbb{R}$ rather than distributions on $\mathbb{R}$, thus reflecting their dependence on where the driving chain is started. We give also necessary and sufficient conditions for the distributional convergence of the forward iterations $Ψ_{n}\circ\ldots\circΨ_{1}$. The main differences caused by the Markovian environment as opposed to the extensively studied case of independent and identically distributed (iid) $Ψ_{1},Ψ_{2},\ldots$ are that: (1) backward iterations may still converge in distribution, if a.s. convergence fails, (2) the degenerate case when $A_{1}c_{M_{1}}+B_{1}=c_{M_{0}}$ a.s. for suitable constants $c_{i}$, $i\in\mathcal{S}$, is by far more complex than the degenerate case for iid $(A_{n},B_{n})$ when $A_{1}c+B_{1}=c$ a.s. for some $c\in\mathbb{R}$, and (3) forward and backward iterations generally have different laws given $M_{0}=i$ for $i\in\mathcal{S}$ so that the former ones need a separate analysis. Our proofs draw on related results for the iid-case, notably by Vervaat, Grincevičius, and Goldie and Maller, in combination with recent results by the authors on fluctuation theory for Markov random walks.

math.PR↗