I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} p(0,t_1,x_1)...p(t_{n-1},t_n,x_n-x_{n-1})dx_1...dx_n$$ then $X$ has independent increments.
I want to use characteristic functions. The idea is that if the characteristic functions factor then this implies independence. So I want to show
$$Ee^{iuX_{t_1}+iv(X_{t_2}-X_{t_1})} = Ee^{iuX_{t_1}}E^{iv(X_{t_2}-X_{t_1})}.$$
I was advised to use Fubini but I am not comfortable with the steps. This is what I have:
$$\int\int e^{iux_1 + iv(x_2-x_1)}p(0,t_1,x_1)p(t_1,t_2,x_2-x_1)dx_1dx_2$$
$$\int\int e^{iux_1 + iv(x_2-x_1)}p(0,t_1,x_1)p(t_1,t_2,x_2-x_1)dx_2dx_1$$
$$\int\int e^{iv(x_2-x_1)}p(t_1,t_2,x_2-x_1)dx_2p(0,t_1,x_1) e^{iux_1}dx_1$$
I once saw someone just push the integral through and end the proof. However, I don't buy this. Maybe there is some weird change of variables? Can someone please justify pushing the integral through and splitting the expected values (if of course it is actually valid.) Thank you
If I make the change of variables $y_2 = x_2-x_1, y_1 = x_1$ then I think I get: $$\int \int e^{ivy_2}p(t_1,t_2,y_2)(dy_2 + dx_1) e^{ivy_1}p(0,t_1,y_1)dy_1$$ which seems to give me an extra term. What am I doing wrong here?
Hint: The substitution $(y_1,y_2)=(x_1,x_2-x_1)$ has Jacobian $1$ hence, for every integrable function $A$, $$ \iint_{\mathbb R^2} A(x_1,x_2-x_1)\,\mathrm dx_1\mathrm dx_2=\iint_{\mathbb R^2} A(y_1,y_2)\,\mathrm dy_1\mathrm dy_2$$