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Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption. (English) Zbl 1430.60042
Summary: Consider a Markov chain $$(X_{n})_{n\geq 0}$$ with values in the state space $$\mathbb{X}$$. Let $$f$$ be a real function on $$\mathbb{X}$$ and set $$S_{n}=\sum_{i=1}^{n}f(X_{i})$$, $$n\geq1$$. Let $$\mathbb{P}_{x}$$ be the probability measure generated by the Markov chain starting at $$X_{0}=x$$. For a starting point $$y\in\mathbb{R}$$, denote by $$\tau_{y}$$ the first moment when the Markov walk $$(y+S_{n})_{n\geq 1}$$ becomes nonpositive. Under the condition that $$S_{n}$$ has zero drift, we find the asymptotics of the probability $$\mathbb{P}_{x}(\tau_{y}>n)$$ and of the conditional law $$\mathbb{P}_{x}(y+S_{n}\leq \cdot\sqrt{n}\mid\tau_{y}>n)$$ as $$n\rightarrow +\infty$$.

##### MSC:
 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems 60J50 Boundary theory for Markov processes 60J05 Discrete-time Markov processes on general state spaces 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60G40 Stopping times; optimal stopping problems; gambling theory
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