FOR CONSISTENT ESTIMATION OF PSD
Let, $x_i(n)$ be a sequence of random variables for $i=1,2, ... , K$ uncorrelated realizations and $\widehat{P}_{(i)}^{per}(e^{jω})$ is their corresponding periodograms. Then, the average of these periodograms is -
$$\widehat{P}_x(e^{jω})=\frac{1}{K}\sum_{i=1}^K\widehat{P}_{(i)}^{per}(e^{jω})$$
Let, $x_i(n)$ be a WSS process also.
Question:
Are the following relations correct?
$E\left\{\widehat{P}_x(e^{jω})\right\}=E\left\{\widehat{P}_{(i)}^{per}(e^{jω})\right\}$ and $Var\left\{\widehat{P}_x(e^{jω})\right\}=\frac{1}{K}Var\left\{\widehat{P}_{(i)}^{per}(e^{jω})\right\}$
I found these two equations in my textbook (Statistical Digital Signal Processing and Modeling, Monson H. Hayes, Page 412).
I am failing to understand these relations (meaning I don't get how you derive them). Can someone please explain them for me?