random walk

Definition. Let $(\mathrm{\Omega },ℱ,𝐏)$ be aprobability space and $\{X_i\}$ a discrete-timestochastic process defined on $(\mathrm{\Omega },ℱ,𝐏)$,such that the $X_i$ are iid real-valued random variables, and$i\in \mathbb{N}$, the set of natural numbers. The randomwalk defined on $X_i$ is the sequence of partial sums, or partialseries

$$S_n:=\sum _{i=1}^n{X}_i.$$

If $X_i\in \{-1,1\}$, then the random walk defined on $X_i$ is called asimple random walk. A symmetric simple random walk isa simple random walk such that $𝐏(X_i=1)=1/2$.

The above defines random walks in one-dimension. One can easilygeneralize to define higher dimensional random walks, by requiringthe $X_i$ to be vector-valued (in ${ℝ}^n$), instead of$ℝ$.

Remarks.

1. Intuitively, a random walk can be viewed as movement in spacewhere the length and the direction of each step are random.

2. It can be shown that, the limiting case of a random walk is aBrownian motion (with some conditions imposed on the $X_i$ so as tosatisfy part of the defining conditions of a Brownian motion). Bylimiting case we mean, loosely speaking, that the lengths of thesteps are very small, approaching 0, while the total lengths of thewalk remains a constant (so that the number of steps is very large,approaching $\mathrm{\infty }$).

3. If the random variables $X_i$ defining the random walk $w_i$are integrable with zero mean $\mathrm{E}[X_i]=0$, $S_i$ is amartingale.