One of the properties of the Wiener process is that its increment is stationary. Which means for time $t$ and $s$ with $0\leq s\leq t$
$$ W_t - W_s \sim W_{t-s} \sim N(0,t-s) $$
This basically says that the difference/Wiener increment is distributed normally with mean $0$ and variance $t-s$. Suppose now I have another sequence of stochastic variables $Y = \{Y_1,Y_2,...,Y_n\}$, what are the rules/conditions that I need to impose, so that $Y$ is also normally distributed and has stationary increments? Namely, $Y$ also satisfies
$$ Y_{t}-Y_{s}\sim Y_{t-s}\sim N(0,t-s) $$
? Naively, I thought that one could simply define such a sequence of stochastic variables $Y$ which maintains such a property. Am I wrong or is there a way to flesh this out rigorously?
The sufficient condition is the next one.
Consider $\xi_i \sim N(0, \sigma_1^2)$, $1 \le i \le n-1$, $\eta \sim N(0, \sigma_2^2)$. Suppose that $(\xi_1, \ldots, \xi_{n-1}, \eta)$ has normal distribution with independent components. Hence $(Y_1, \ldots, Y_n) = (\eta, \eta+\xi_1, \eta+\xi_1+\xi_2, \ldots, \eta+\xi_1 + \ldots + \xi_{n-1})$ has normal distribution and stationary increments.