I am reading Brownian Motion and Stochastic Calculus by Karatzas and Shreve and am having trouble proving a statement made near equation (2.5).
Let $(\Omega,\mathcal{F},P)$be a probability space. For any stochastic process $X : \Omega \times \mathbb{R} \to \mathbb{R}$ let $X_t(\omega) := X(\omega,t)$.
Given a finite set of non-negative real numbers $(s_1,s_2,\ldots,s_n)$ where $n$ is any natural number, if the cumulative distribution function of $X_{s_1},X_{s_2},\ldots,X_{s_n}$ is \begin{align*} F_{(s_1,s_2,\ldots,s_n)}(z_1,z_2,\ldots,z_n) = \int_{-\infty}^{z_1} \ldots \int_{-\infty}^{z_n} p(s_1;0,y_1)p(s_2-s_1;y_2,y_1)\ldots p(s_n-s_{n-1};y_{n-1},y_n) dy_n \ldots dy_1 \end{align*} where \begin{align*} p(q;r,s) := \frac{1}{\sqrt{2\pi q}}\exp\left( -\frac{|r-s|^2}{2q} \right) \end{align*}
then prove that $X$ has independent increments which are normally distributed with zero mean and variance as difference of time index.
Any hints or proofs are appreciated.
What I have tried:
Where I am stuck is I do not know how to begin thinking of indepencence. I know that to show independence, I can show that $$P(X_{s_1} \le z_1, X_{s_2} - X_{s_1} \le z_2) = P(X_{s_1} \le z_1)P(X_{s_2} - X_{s_1} \le z_2),$$ but how do I calculate the right and left hand side here?