Explanation

Random walks are a special case of an autoregressive model of order one (AR(1)).

Key points:

• A random walk model is defined as: $x_t = x_{t-1} + \varepsilon_t$
• This can be rewritten as: $x_t - x_{t-1} = \varepsilon_t$
• This is equivalent to an AR(1) model where the coefficient on the lagged term is exactly 1
• In AR(1) notation: $x_t = \phi x_{t-1} + \varepsilon_t$ where $\phi = 1$

Therefore, random walks are a specific instance of AR(1) models where the autoregressive coefficient equals 1.