Autocovariància de processos estocàstics Amb la notació habitual E {\displaystyle \operatorname {E} } d'expectativa , si el procés estocàstic { X t } {\displaystyle \left\{X_{t}\right\}} mitjana μ t = E [ X t ] {\displaystyle \mu _{t}=\operatorname {E} [X_{t}]} [3]
K X X ( t 1 , t 2 ) = cov [ X t 1 , X t 2 ] = E [ ( X t 1 − μ t 1 ) ( X t 2 − μ t 2 ) ] = E [ X t 1 X t 2 ] − μ t 1 μ t 2 {\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {cov} \left[X_{t_{1}},X_{t_{2}}\right]=\operatorname {E} [(X_{t_{1}}-\mu _{t_{1}})(X_{t_{2}}-\mu _{t_{2}})]=\operatorname {E} [X_{t_{1}}X_{t_{2}}]-\mu _{t_{1}}\mu _{t_{2}}}
()
on t 1 {\displaystyle t_{1}} t 2 {\displaystyle t_{2}}
Propietat de simetria K X X ( t 1 , t 2 ) = K X X ( t 2 , t 1 ) ¯ {\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})={\overline {\operatorname {K} _{XX}(t_{2},t_{1})}}}
respectivament per a un procés WSS:
K X X ( τ ) = K X X ( − τ ) ¯ {\displaystyle \operatorname {K} _{XX}(\tau )={\overline {\operatorname {K} _{XX}(-\tau )}}}
Filtrat lineal L'autocovariància d'un procés filtrat linealment { Y t } {\displaystyle \left\{Y_{t}\right\}}
Y t = ∑ k = − ∞ ∞ a k X t + k {\displaystyle Y_{t}=\sum _{k=-\infty }^{\infty }a_{k}X_{t+k}\,}
és
K Y Y ( τ ) = ∑ k , l = − ∞ ∞ a k a l K X X ( τ + k − l ) . {\displaystyle K_{YY}(\tau )=\sum _{k,l=-\infty }^{\infty }a_{k}a_{l}K_{XX}(\tau +k-l).\,}
Referències 
i
són dos casos en el temps.
![{\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {cov} \left[X_{t_{1}},X_{t_{2}}\right]=\operatorname {E} [(X_{t_{1}}-\mu _{t_{1}})(X_{t_{2}}-\mu _{t_{2}})]=\operatorname {E} [X_{t_{1}}X_{t_{2}}]-\mu _{t_{1}}\mu _{t_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c3c2da09a072e13e513b1bbe6933e25cde5645)
