 # Cross-correlation

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In probability and statistics, the term cross-correlations refers to the correlations between the entries of two random vectors $\mathbf {X}$ and $\mathbf {Y}$ , while the correlations of a random vector $\mathbf {X}$ are the correlations between the entries of $\mathbf {X}$ itself, those forming the correlation matrix of $\mathbf {X}$ . If each of $\mathbf {X}$ and $\mathbf {Y}$ is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of $\mathbf {X}$ are known as autocorrelations of $\mathbf {X}$ , and the cross-correlations of $\mathbf {X}$ with $\mathbf {Y}$ across time are temporal cross-correlations. In probability and statistics, the definition of correlation always includes a standardising factor in such a way that correlations have values between −1 and +1.
If $X$ and $Y$ are two independent random variables with probability density functions $f$ and $g$ , respectively, then the probability density of the difference $Y-X$ is formally given by the cross-correlation (in the signal-processing sense) $f\star g$ ; however, this terminology is not used in probability and statistics. In contrast, the convolution $f*g$ (equivalent to the cross-correlation of $f(t)$ and $g(-t)$ ) gives the probability density function of the sum $X+Y$ .