Zeitkorrelierte

Zeitkorrelierte, or time-correlated, is a term used in physics, statistics and signal processing to describe stochastic quantities whose values at different times are statistically linked. In a zeitkorrelierte process the joint distribution of X(t) and X(t+τ) depends on the time lag τ. By contrast, white noise is (approximately) time-uncorrelated, with observations at distinct times independent of one another.

The degree of time correlation is formalized by the autocorrelation function R(τ) = E[(X(t) - μ)(X(t+τ) - μ)]. If R(τ) decays slowly, the process retains memory; the characteristic decay time is called the correlation time τ_c. Common examples include the Ornstein–Uhlenbeck process, which has exponential decay R(τ) ∝ e^{-|τ|/τ_c}, and AR(1) time series with R(τ) ∝ ρ^{|τ|}. Colored noises such as pink noise (1/f) and Brownian noise (1/f^2) exhibit significant temporal structure over a range of τ.

Applications include physics, where time correlation reflects relaxation processes, and climate science, finance, and engineering, where many signals are not white noise. In communications and sensing, temporal correlations affect detection, estimation and filtering; neglecting them can bias results. Modeling approaches include autoregressive moving-average (ARMA/ARIMA) models and Gaussian processes with kernels that encode the desired correlation.

Analysis methods focus on estimating the autocorrelation or the power spectral density to characterize the correlation structure. Practitioners may whiten data, use robust estimators, or explicitly model the dependence in inference. Zeitkorrelierte phenomena are central to the study of stochastic processes, time series analysis and spectral theory.