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Homerandom walk

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# random walk

Definition. Let $(\Omega,\mathcal{F},\mathbf{P})$ be a
probability space and $\{X_{i}\}$ a discrete-time
stochastic process defined on $(\Omega,\mathcal{F},\mathbf{P})$,
such that the $X_{i}$ are iid real-valued random variables, and
$i\in\mathbb{N}$, the set of natural numbers. The *random
walk* defined on $X_{i}$ is the sequence of partial sums, or partial
series

$S_{n}\colon=\sum_{{i=1}}^{{n}}X_{i}.$ |

If $X_{i}\in\{-1,1\}$, then the random walk defined on $X_{i}$ is called a
*simple random walk*. A *symmetric simple random walk* is
a simple random walk such that $\mathbf{P}(X_{i}=1)=1/2$.

The above defines random walks in one-dimension. One can easily generalize to define higher dimensional random walks, by requiring the $X_{i}$ to be vector-valued (in $\mathbb{R}^{n}$), instead of $\mathbb{R}$.

Remarks.

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

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

3. If the random variables $X_{i}$ defining the random walk $w_{i}$ are integrable with zero mean $\operatorname{E}[X_{i}]=0$, $S_{i}$ is a martingale.

## Mathematics Subject Classification

60G50*no label found*82B41

*no label found*

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