Autocovariance and sample autocovariance

For a weakly stationary real process $X=(X_t{)}_{t\in \mathbb{Z}}$, its autocovariance function (ACVF) is $\gamma (h):=\mathrm{Cov}(X_{t+h},X_t),h\in \mathbb{Z},$ which is independent of $t$ by Strict and weak stationarity for integer-indexed time series.

For a finite data vector $x_1,\dots ,x_n\in \mathbb{R}$, where $n\ge 1$, let ${\bar{x}}_n:=\frac{1}{n}{\sum }_{t=1}^n{x}_t\mathrm{.}$ Its biased sample ACVF is the zero-extended sequence on $\mathbb{Z}$ given by ${\widehat{\gamma }}_n(h):=\frac{1}{n}{\sum }_{t=1}^{n-\mathrm{\mid }h\mathrm{\mid }}(x_{t+\mathrm{\mid }h\mathrm{\mid }}-{\bar{x}}_n)(x_t-{\bar{x}}_n),\mathrm{\mid }h\mathrm{\mid }<n,$ and ${\widehat{\gamma }}_n(h):=0$ for $\mathrm{\mid }h\mathrm{\mid }\ge n$. The absolute lag makes ${\widehat{\gamma }}_n(-h)={\widehat{\gamma }}_n(h)$.

For comparison, define the pair-count-normalized rescaling ${\tilde{\gamma }}_n(h):=\frac{1}{n-\mathrm{\mid }h\mathrm{\mid }}{\sum }_{t=1}^{n-\mathrm{\mid }h\mathrm{\mid }}(x_{t+\mathrm{\mid }h\mathrm{\mid }}-{\bar{x}}_n)(x_t-{\bar{x}}_n),\mathrm{\mid }h\mathrm{\mid }<n,$ and set it to zero for $\mathrm{\mid }h\mathrm{\mid }\ge n$. Its denominator is positive on the stated domain. The names specify the denominator convention; they assert no unbiasedness formula.