Suppose there are two time series. The first one is $x(t)$ where $t$ is the time. The second one is $y(t)=\int_{-\infty}^{+\infty} x(s)Q(t-s)\mathrm{d} s$ where $Q(s)$ is a weighting function and $\int_{-\infty}^{+\infty} Q(s)\mathrm{d} s=1$. The autocorrealtion function is $\rho_x(\tau)$ for $x(t)$ and $\rho_y(\tau)$ for $y(t)$.
I want to prove that $\rho_y(\tau)\gt\rho_x(\tau)$. It is very intuitive that $\rho_y(\tau)\gt\rho_x(\tau)$ because $y(t)$ is the result of smoothing of $x(t)$. Smoothing or averaging can increase the autocorrelation. However, I cannot mathmatically prove that. At least no success since this morning.
Can anyone please help prove that $\rho_y(\tau)\gt\rho_x(\tau)$?
Thank you