Let $\{W^{(1)}_t , t\ge0 \} $ and $\{W^{(2)}_t , t\ge0 \} $ be independent Wiener processes. Find all constants for which $\alpha W^{(1)}_t +\beta W^{(2)}_t, t\ge0$ is Wiener process.
I already saw the answers to the questions
But did not understand how they were resolved. I will be grateful for any advice.
$W_t := \alpha W_t^{(1)} + \beta W_t^{(2)}$. The covariance function of a Wiener process is $t \wedge s$ (and it is unique for gaussian processes). Thus we want to solve $$\Bbb E [W_t W_s] \overset{!}{=} t\wedge s$$ The left hand side can be transformed to $\alpha^2 (t\wedge s) + \beta^2 (t\wedge s)$. Thus $W_t$ is a Wiener process iff $\alpha^2 + \beta^2 = 1$
