Random walk vs AR(1)
One AR(1) process, yₜ = φ·yₜ₋₁ + eₜ with y₀ = 0 and the same fixed shocks eₜ. Slide the persistence φ: for |φ| < 1 the path is mean-reverting to 0, at φ = 1 it is a random walk (unit root) whose shocks never fade.
φ = 0.50Stationary (mean-reverting)
Path ends at y₁₂₀ −0.41Var(yₜ) at φ = 1, t = 120 120 = t·σₑ²
Persistence φ0.50
With φ = 0.50 the process is Stationary (mean-reverting): shocks fade geometrically, so the path keeps returning to its mean of 0. The smaller φ is, the faster the pull back to 0. Raise φ toward 1 and the reversion gets weaker, so the path strays further before returning.