I have this random process given by a random walk, that is, knowing the position at a given time X[t], the next position will be
X[t+1] = X[t] + V
where V is a random variable with Gaussian distribution, with mean value 0 and variance ², and the initial position is X[0] = x0. And I want to determine if this process is wide sense stationary (WSS) or not. For that, I know I need to calculate its mean value (for which I got x0) and its autocorrelation.
However, I am struggling to find its autocorrelation (that is E{X[t]X[t + Δt]}). I know it should be the first integral below, but I cant solve it and dont know if the equality is right. Could someone please help me find a way to go?
$\int_{-\infty}^\infty X[t]X[t + Δt] dt = \int_{-\infty}^\infty [x_0+V][x_0+V]dt$
Many thanks.