No abstract

a b s t r a c tWe consider the equation R n = Q n + M n R n−1 , with random non-i.i.d. coefficients (Q n , M n ) n∈Z ∈ R 2 , and show that the distribution tails of the stationary solution to this equation are regularly varying at infinity.

The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE). Then, we prove that the corresponding ODE converges to local optima and stays there. Consequently, we conclude that the cGA will converge to the local optima of the function to be optimized.

a b s t r a c tWe study a linear recursion with random Markov-dependent coefficients. In a ''regular variation in, regular variation out'' setup we show that its stationary solution has a multivariate regularly varying distribution. This extends results previously established for i.i.d. coefficients.

We consider a generalized version of a directionally reinforced random walk, which was originally introduced by Mauldin, Monticino, and von Weizsäcker in [17]. Our main result is a stable limit theorem for the position of the random walk in higher dimensions. This extends a result of Horváth and Shao [11] that was previously obtained in dimension one only (however, in a more stringent functional form). MSC2000: Primary: 60F15, 60F17, 60F20; Secondary: 60J25, 70B05.