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For an additive time series $X_t=T_t+S_t+{\epsilon }_t$ with period-$m$ seasonal component and ${\sum }_{j=1}^m{S}_j=0$, a centred average spanning one complete period removes the seasonal component while estimating the slowly varying trend. When $m$ is even, average two adjacent length-$m$ averages to centre the filter on an observation; the endpoints receive half the usual weights. Estimate each seasonal phase by averaging the corresponding detrended residuals across cycles, then subtract the mean of those phase averages to enforce the zero-sum convention. Multiplicative seasonality instead uses ratios and a mean-one convention. Repeated averaging is filter composition and induces serial dependence even from white noise.

A time series is a sequence of observations indexed by time: hourly temperatures, daily sales, or successive measurements of a fluctuating signal. A probabilistic model treats these observations as one sample path of a stochastic process $(X_t)$. For equally spaced observations, take $t\in \mathbb{Z}$ and measure a lag of a time series in observation intervals. The order matters: shuffling a record preserves its observed values but destroys information about how one value follows another.

Two basic ways to model temporal dependence are an autoregressive model, which feeds past observations into the next value, and a moving-average model, which combines present and past white noise. An AR(1) carries a disturbance forward with diminishing weights; an MA(1) carries it into just two observations. Their autocorrelation functions make this difference visible. A weakly stationary process supplies a common mean and a lag-dependent autocovariance with which to describe that dependence.

These models help forecast future observations and quantify the uncertainty of averages. Successive measurements need not provide as much information as the same number of independent random variables: the long-run variance of a stationary process accounts for their accumulated covariances. Trend and seasonality require attention before fitting a stationary model; differencing and a seasonal difference operator can help, but do not automatically make every process stationary. See Peter Bartlett, Introduction to Time Series Analysis, lecture 1: objectives and models.

The three illustrated sample paths have zero mean and unit variance, so the visual comparison concerns dependence rather than scale. The AR(1) paths start from their stationary normal distribution, independently of future discrete Gaussian white noise; the MA(1) path includes the noise preceding its first observation. These are finite random examples, not exact pictures of an ACF. Compare their theoretical autocorrelation functions in autocorrelation.

Historically, Yule's 1927 study investigated disturbed periodicities in sunspot observations; it should not be read as the discovery of the particular AR(1) examples here. Slutsky studied how summing random inputs could produce wave-like records, a warning against inferring a periodic cause from appearance alone. The editorial note to his 1937 English paper identifies the 1927 Russian original. See G. Udny Yule, On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers (1927) and Eugen Slutsky, The Summation of Random Causes as the Source of Cyclic Processes (1937), including the editorial note on the 1927 original.

The unnormalized autocorrelation function of a stationary velocity. For zero mean, $\chi (0)$ is the velocity variance. It is different from the normalized autocorrelation function $\chi (u)/\chi (0)$ often used in time series.

If $X(t)-X(0)={\int }_0^{t}V(s)ds$, integrating the two-time covariance over the square $[0,t{]}^2$ givesThis identity connects a mean squared displacement to the Green-Kubo formula for diffusion.

A second-order stationary time series decomposes into a deterministic component and a one-sided square-summable linear filter of a stationary time series of its mutually orthogonal linear innovations. Normalize $c_0=1$. For a causal invertible ARMA process with no deterministic component, the driving white noise is this innovation process: observations and innovations span the same closed past subspace, while the next innovation is orthogonal to it.