# Piecewise Linear Probability Density Functions of Invariant Measures for Position Dependent Random Maps

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Let the map f:[âˆ’1,1]â†’[âˆ’1,1] have a.c.i.m. Ï (absolutely continuous f-invariant measure with respect to Lebesgue). Let Î´ Ï be the change of Ï corresponding to a perturbation X= Î´ fâ—‹ fâˆ’1 of f. Formally we have, for differentiable A, but this...

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Abstract. Let {X[sub n]}[sub n is greater than or equal to 0] be a Harris recurrent Markov chain with state space E and invariant measure pi. The law of the iterated logarithm and the law of weak convergence are given for the additive functionals of the form [Multiple line equation(s) cannot be...

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Let Yk (?) (k ? 0) be the number of vertices of a Galton-Watson tree ? that have k children, so that Z(?) := Sk?0 Yk(?) is the total progeny of ?. In this paper, we will prove various statistical properties of Z and Yk. We first show, under a mild condition, an asymptotic expansion of P(Z = n)...

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Not much is known about the general properties of nonlinear time discrete periodic diffusion networks. Therefore it is important to discuss a specific network called the Four-Number Game which is historically introduced in the form of a mathematical curiosity. In our network, the dynamical...

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For a family of real-valued Gaussian processes ? u ( t), t ? [0, T], we obtain an exact asymptotics of the probability of crossing a level u as u ? 8 under certain conditions on the variance and correlation. This result is applied to the investigation of excursions of a stationary zero-mean...