The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space it is true that $$ |\langle x,y\rangle| ^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle$$  where \( \langle\cdot,\cdot\rangle \) is the inner product, also known as dot product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as$$ |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.$$

The cross-covariance function scaled to live in [-1, 1] is called cross-correlation function and given by$$ \rho_{xy} (s,t)=\frac{ \gamma_{xy}(s,t)}{\sqrt{\gamma_x(s,s)\gamma_y(t,t)}}. $$

The cross-covariance function between two series, \( x_t \) and \( y_t \) , is$$  \gamma_{xy} (s,t)=cov(x_s,y_t)=E[(x_s- \mu_{xs})(y_t- \mu_{yt})].$$