AutoKovarianz

AutoKovarianz, in English known as autocovariance, is a function that measures the dependency structure of a stochastic process byCov(X_t, X_{t+k}). For a weakly stationary process with constant mean mu, it is defined as gamma(k) = Cov(X_t, X_{t+k}) = E[(X_t - mu)(X_{t+k} - mu)]. The function depends only on the lag k. The value gamma(0) equals the variance of the process, Var(X_t) = sigma^2. The autocovariance function is even: gamma(-k) = gamma(k). The autocovariance is related to the autocorrelation function by rho(k) = gamma(k) / gamma(0).

Estimation of AutoKovarianz from data proceeds with the sample autocovariance. For a time series x_1, ..., x_n, the estimator at lag h is gamma_hat(h) = (1/(n - h)) sum_{t=1}^{n-h} (x_t - x_bar)(x_{t+h} - x_bar), where x_bar is the sample mean. If the population mean mu is known, mu can replace x_bar. This estimator is consistent as n grows, but it is biased for finite samples when mu is unknown; the bias vanishes asymptotically.

The autocovariance function connects to the spectral density f(λ) via f(λ) = (1/2π) ∑_{k=-∞}^{∞} gamma(k) e^{-i k λ}. In common models, AR(1) processes yield gamma(k) = φ^{|k|} Var(X_t), while MA(q) processes have nonzero gamma(k) only for |k| ≤ q.

Applications of AutoKovarianz include characterizing dependence, informing model identification (as in ARMA modeling), and underpinning spectral analysis and statistical inference for time series.