Is maximum cross-correlation achieved at the origin?

Question

Let $x[n]$ and $y[n]$ be two DT random signals with $x[n]\xrightarrow{\mathcal{H}}y[n]$ through some system $\mathcal{H}$ that is deterministic yet unknown.\ Define both the autocorrelation and cross-correlation of the signals: \begin{align*} R_{x}[k]&\triangleq\langle x[n]x[n-k]\rangle\tag{1}\\ R_{xy}[k]&\triangleq\langle x[n]y[n+k]\rangle\tag{2} \end{align*} A major property of $(1)$ is that $R_{x}[k]$ is maximum at the origin (by origin I mean origin of $R_{x}[k]$) i.e., $k=0$. A property that can be proven by Cauchy-Schwartz is that : $$ |R_{xy}[k]|\leq R_{x}[0]R_{y}[0]\tag{3} $$ meaning that the maximum of the cross correlation is achieved when $R_{x}[k]$ is measured at $k=0$ and when $R_{y}[k]$ is measured at $k=0$.

My Question: Does this imply $R_{xy}[k]$ is maximum at $k=0$? because it seems that at this value we have by definition $R_{xy}[0]=\langle x[n]y[n]\rangle\neq R_{x}[0]R_{y}[0]$. Therefore, is there a possibility that the maximum of $R_{xy}[k]$ occur at $k\neq0$?

Answer 1

meaning that the maximum of the cross correlation is achieved when $R_{x}[k]$ is measured at $k=0$ and when $R_{y}[k]$ is measured at $k=0$.

Nope. It just means that the absolute maximum of the cross correlation is smaller than the product of the absolute maxima of the auto correlations of the individual signals. It doesn't say anything about where the absolute maximum of the cross correlation occurs in time (or lag). I.e.

$$ max(|R_{xy}[k]|) < R_x[0] \cdot R_y[0]$$

My Question: Does this imply $R_{xy}[k]$ is maximum at $k=0$?

No

Therefore, is there a possibility that the maximum of $R_{xy}[k]$ occur at $k\neq0$?

Absolutely. Just look at a simple example of two unit impulses that are shifted in time, i.e. $x[k] = \delta [k]$ and $y[k] = \delta [k-M]$ The cross correlation will have it's maximum at $k = -M$.