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The labels in PDF Figure 2 match as follows:Series $1$ has a wandering level and an overall upward drift. Its empirical autocorrelation function stays close to $1$ and declines very slowly, matching panel $c$; this is characteristic of a trend or a record from a unit-root autoregressive process. Such an empirical ACF is not a population stationary ACF if the underlying model is a random walk.

Series $2$ shows relatively smooth runs of adjacent observations on the same side of its level. Positive short-lag dependence emphasizes low frequencies, matching the decreasing spectrum $d$.

Series $3$ is rougher, with much more rapid fluctuation. Negative short-lag dependence emphasizes high frequencies, matching the increasing spectrum $a$. A negative-coefficient AR(1), for example, has this type of spectrum; the figure does not uniquely identify its exact order or parameters.

Series $4$ has conspicuous seasonality. Its autocorrelation function should alternate between positive peaks at full seasonal periods and negative troughs between them, matching $b$. Here “characteristic function” is used descriptively for an ACF or spectrum, rather than for the probability-theoretic characteristic function of a random variable.

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.