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Limit theorems for Markov walk conditioned to stay positive

Abstract : We consider a real random walk whose increments are constructed by a Markov chain definedon an abstract space. We suppose that the random walk is centred and that the dependence of the Markov walk in its past decreases exponentially fast (due to the spectral gap property). We study the first time when the random walk exits the positive half-line and prove that the asymptotic behaviour of the survey probability is inversely proportional to the square root of the time. We extend also to our Markovian model the following result of random walks with independent increments: the asymptotic law of the random walk renormalized and conditioned to stay positive is the Rayleigh law. Subsequently, we restrict our model to the cases when the Markov chain defining the increments of the random walk takes its values on a finite state space. Under this assumption and the condition that the walk is non-lattice, we complete our results giving local theorems for the random walk conditioned to stay positive. Finally, we apply these developments to branching processes under a random environment defined by a Markov chain taking its values on a finite state space. We give the asymptotic behaviour of the survey probability of the process in the critical case and the three subcritical cases (strongly, intermediate and weakly).
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Ronan Lauvergnat. Limit theorems for Markov walk conditioned to stay positive. Probability [math.PR]. Université de Bretagne Sud, 2017. English. ⟨NNT : 2017LORIS451⟩. ⟨tel-01718390⟩

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