Our goal is to investigate the asymptotic behavior of center mass elephant random walk, which a discrete-time walk on integers with complete memory its whole history. In diffusive and critical regimes, we establish almost sure convergence, law iterated logarithm quadratic strong for walk. The normality, properly normalized, also provided. Finally, prove limit theorem in superdiffusive regime. A...

Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation ...

We introduce a multi-coin discrete quantum walk where the amplitude for a coin flip depends upon previous tosses. Although the corresponding classical random walk is unbiased, a bias can be introduced into the quantum walk by varying the history dependence. By mixing the biased walk with an unbiased one, the direction of the bias can be reversed leading to a new quantum version of Parrondo’s pa...

We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum random walks. We provide the framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We t...

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time n. Assuming that the moment of order 2 + δ is finite, we show that the exact asymptotics for this probability are n−1/4. To show these asymptotics we develop a discrete potential theory for integrated random walks.