# Positive definites of correlation functions

Say I you have two time series, $x_k, k=1,2$ generated from two, possibly correlated, complex gaussian processes. The lag 0 and 1 auto-correlation estimate for the two time series is denoted $R_{0,k}, R_{1,k}, k=1,2$.

The normalized auto correlation is denoted $\rho_k = \frac{R_{1,k}}{R_{0,k}}$, with the following property $|\rho_k| \leq 1$.

If you compute the weighted average of $R_0$ and $R_1$ separetely, i.e.

$$\bar{R_{0}} = \alpha_1R_{0,1} + \alpha_2R_{0,2} \\ \bar{R_{1}} = \alpha_1R_{1,1} + \alpha_2R_{1,2} \\ \alpha_1 + \alpha_2 = 1,$$

can one prove that $\bar{\rho} = \frac{\bar{R_1}}{\bar{R_0}} \leq 1$ or prove that this is not the case?

Calculating the weighted average of $\rho_{1,2}$ is $\leq 1$.

If the two time series are uncorrelated, the average of the correlation functions is positive definite, since the average of the auto correlation functions is the same as the autocorrelation of the average of the two time series.

• 1) Autocorrelation is cross correlation of a signal $x$ with itself. 2) Auto-correlation is already normalised 3) The subscript on $\rho$ doesn't make any sense. 4) What are you trying to do? What is the application? What is the objective? What is the motivation behind the question? – A_A Aug 13 '18 at 10:53