The foundation of time series analysis is stationarity. A time series {rt} is said to be strictly stationary if the joint distribution of 
is identical to that of 
for all t, where k is an arbitrary positive integer and (t1, … , tk) is a collection of k positive integers. In other words, strict stationarity requires that the joint distribution of 
is invariant under time shift. This is a very strong condition that is hard to verify empirically. A weaker version of stationarity is often assumed. A time series {rt} is weakly stationary if both the mean of rt and the covariance between rt and rt−ℓ are time invariant, where ℓ is an arbitrary integer. More specifically, {rt} is weakly stationary if (a) E(rt) = μ, which is a constant, and (b) Cov(rt, rt−ℓ) = γℓ, which only depends on ℓ. In practice, suppose that we have observed T data points {rt|t = 1, … , T}. The weak stationarity implies that the time plot of the data would show that the T values fluctuate with constant variation around a fixed level. In applications, weak stationarity enables one to make inference concerning future observations (e.g., prediction).
Implicitly, in the condition of weak stationarity, we assume that the first two moments of rt are finite. From the definitions, if rt is strictly stationary and its first two moments are finite, then rt is also weakly stationary. The converse is not true in general. However, if the time series rt is normally distributed, then weak stationarity is equivalent to strict stationarity. In this book, we are mainly concerned with weakly stationary series.
The covariance γℓ = Cov(rt, rt−ℓ) is called the lag-ℓ autocovariance of rt. It has two important properties: (a) γ0 = Var(rt) and (b) γ−ℓ = γℓ. The second property holds because Cov(rt, rt−(−ℓ)) = Cov(rt−(−ℓ), rt) = Cov(rt+ℓ, rt) = Cov(rt1, rt1−ℓ), where t1 = t + ℓ.
In the finance literature, it is common to assume that an asset return series is weakly stationary. This assumption can be checked empirically provided that a sufficient number of historical returns are available. For example, one can divide the data into subsamples and check the consistency of the results obtained across the subsamples.