4 Random Walk

Let \(\{Z_t\}\) be a purely random process with mean \(\mu \), variance \(\sigma ^2\). Then the process \(\{X_t\}\) is called a random walk if \(X_t = X_{t-1} + Z_t\). We usually start at \(t = 1\), so that \(X_1 = Z_t\) and \(X_t = \sum \limits ^t_{i=1}Z_i\) , in this regard \(E(X_t) = t\mu \) and \(var(X_t) = t\sigma ^2\)

Since mean and variance change with time the process is non-stationary. However the difference of \(X_t-X_{t-1}\) is purely random
i.e \(\nabla X_t = X_t - X_{t-1} = Z_t\).

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