Random Walk

Definition

Let $\{e_t\}$ : White noise

A random walk process is defined as:

$$\begin{array}{rl}{Y}_1& =e_1\\ Y_t& =Y_{t-1}+e_t\end{array}$$

It can also be written as:

$$Y_t=\sum _{i=1}^t{e}_i$$

Tip

It is AR(1) with $\varphi =1$

Variance

$\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y)$ .

White noise is independent, and identically distributed. So, $\text{Cov}(e_s,e_t)=0$, and $\text{Var}(e_s+e_t)=\text{Var}(e_s)+\text{Var}(e_t)=2{\sigma }_e^2$ .

From this property we can obtain

$$\begin{array}{rl}\text{Var}(Y_t)& =\text{Var}\left(\sum _{i=1}^t{e}_i\right)\\ & =t\cdot {\sigma }_e^2\end{array}$$

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