Let {X(t)} and {Y(t)} denote the input and output respectively.
If h(t) is the system weighing function, then
Y(t)= $∫^\infty_{-\infty} h(β). X(t-β) dβ
Autocorrelation of output is given as
$R_{YY}$($t_1$,$t_2$) = E{Y($t_1$.Y*($t_2$)}
=E$∫^\infty_{-\infty} h(β). X({t_1}-β) dβ {Y^*}({t_2})$
=$∫^\infty_{-\infty} E{X({t_1}-β){Y^*}({t_2})} h(β).dβ$
= $∫^\infty_{-\infty} {R_{XY}}({t_1}-β,{t_2})h(β).dβ$
=$R_{XY} ({t_1},{t_2})*h({t_1})$
Thus the autocorrelation of output is given by convolution of cross-correlation( between input-output ) and LTI system impulse response.