Autocorrelation C(τ) = ⟨x(t)x(t+τ)⟩/⟨x²⟩ measures how correlated a signal is with its future self. Its decay rate characterizes the memory in a stochastic process.
OU: C(τ) = exp(−τ/θ) | AR(1): C(k) = φ^k | Long memory: C(τ) ~ τ^(2H−2)
Short-memory processes (OU, AR(1)) have exponential decay — the correlation length is finite. Long-memory (fractional) processes with Hurst exponent H > 1/2 have power-law decay with divergent correlation time, relevant to climate, finance, and turbulence.