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 $k\neq0$?