Random Walk and White Noise

On the lecture slides the definition of a random walk is given as

$$\begin{aligned} y_t &= y_{t-1} + u_t, \quad &\text{for}\: t = 1,2,\ldots \quad &(\text{random walk without drift}) \\ y_t &= \mu + y_{t-1} + u_t, \quad &\text{for}\: t = 1,2,\ldots \quad &(\text{with drift parameter } \mu), \end{aligned}$$

with white noise \(u_t\), which means \(E(u_t) = 0\) and \(Var(u_t) = \sigma_u^2\) and \(Corr(u_t,u_s) = 0 \) for \(t \neq s\), and assuming that the initial value \(y_0\) equals 0.

The two equations above can be rewritten as a cumulative sum of white noise variates. First, rewrite the equation for a random walk without drift,

$$\begin{aligned} y_t &= y_{t-1} + u_t \\ y_t &= y_{t-2} + u_{t-1} + u_t \\ y_t &= y_{t-3} + u_{t-2} + u_{t-1} + u_t \\ \vdots \\ y_t &= \sum\limits_{j = 1}^{j = t} u_j, \end{aligned}$$

and for a random walk with drift parameter \(\mu\),

$$\begin{aligned} y_t &= \mu + y_{t-1} + u_t \\ y_t &= \mu + \mu + y_{t-2} + u_{t-1} + u_t \\ y_t &= \mu + \mu + \mu + y_{t-3} + u_{t-2} + u_{t-1} + u_t \\ \vdots \\ y_t &= \mu t + \sum\limits_{j = 1}^{j = t} u_j. \end{aligned}$$

To generate a gaussian random walk (with and without drift) in R we need the functions rnorm(), set.seed(), and cumsum().