### A central limit theorem for random walks in random labyrinths

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We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.

In this paper we study the almost sure conditional central limit theorem in its functional form for a class of random variables satisfying a projective criterion. Applications to strongly mixing processes and nonirreducible Markov chains are given. The proofs are based on the normal approximation of double indexed martingale-like sequences, an approach which has interest in itself.

In this paper we obtain a strong invariance principle for negatively associated random fields, under the assumptions that the field has a finite $(2+\delta )$th moment and the covariance coefficient $u\left(n\right)$ exponentially decreases to $0$. The main tools are the Berkes-Morrow multi-parameter blocking technique and the Csörgő-Révész quantile transform method.

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.

We prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables with finite second moment.