Abstract

In this talk, starting from the sum of independent and identically distributed (i.i.d.) random variables, I will introduce the classical simple random walk along with its limit behaviors. After that, some extensions will be made in several directions. Firstly, consider the nearest neighbor random walk with non-homogeneous tran- sition probabilities. By calculating the escaping probabilities, the criteria for recur- rence will be given.

Secondly, by perturbing the recurrent simple random walk, I will study the near- recurrent random walk, which is known as the “Lamperti Problem”. The recurrence criteria and the cut points will be discussed.

At last, by randomizing the transition probabilities (e.g., let the transition proba- bilities be i.i.d. ), I will introduce the so-called random walk in random environment (RWRE), which is a very hot research object in the community of probability s- tudies. RWRE exhibits a lot of interesting properties. For example, it may run much slower than the simple random walks. Precisely, it’s possible that the random walk is transient to the infinity but the speed is zero, which is impossible in any circumstances for simple random walks.

This talk only involves some basic knowledge of the elementary probability such as the law of large numbers, the mathematical expectation of random variables, etc.