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We consider a random walk $tilde S$ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that ${frac{1}{sqrt n} tilde S(nt)}$ has no weak limit in $De$; alternatively, the weak limit is a reflected Brownian motion.
We consider a simple random walk on $mathbb{Z}^d$ started at the origin and stopped on its first exit time from $(-L,L)^d cap mathbb{Z}^d$. Write $L$ in the form $L = m N$ with $m = m(N)$ and $N$ an integer going to infinity in such a way that $L^2 s
We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depe
We consider a random walk with a negative drift and with a jump distribution which under Cramers change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this
A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A ra
We work under the A{i}d{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown t