# Why does the residual and variable cross correlation function tell us about cross independence?

I am new to time series. I understand that the sample autocorrelation function (SACF), if Xt is white noise, then for large (n), the SACF is approximately normally distributed with zero mean and standard deviation 1/sqrt(n). For Yt = signal + white noise, if we make a model and subtract it from Yt. If the residuals are close to white noise by using 1/sqrt(n) hypothesis test, we can conclude that the model is good.

The confusing thing to me is this: It works only if Xt and Yt are independent linear processes then the large sample distribution of sample cross correlation function has mean 0 and standard deviation 1/sqrt(n).

If we use these equations: There is strong seasonality.  The removal of signal from Yt: Why does the cross correlation function of the residual of yt and Xt tell you that Xt and Yt are cross independence? Shouldn't Xt and Yt tell you directly?

Yw is the residual: Thanks