Suppose $Y,Z$ are i.i.d. random variables, obeying the standard normal distribution, and let $$X(t)=Y \cos(\theta t) + Z \sin(\theta t)$$for $\forall t \in \mathcal{R}$ and a fixed $\theta$. Then $X_t(Y,Z)$ is a stochastic process.
Then what is the finite dimensional distribution of $X_t$? That is, for any given k moments $t_1,t_2, \cdots,t_k$, what is the joint distribution $P(X_{t_1}=x_1,\cdots,X_{t_k}=x_k)$ ?
What I've done :
- It is not hard to show that for a fixed $t$, $X(t) \sim N(0,1)$.
- I wonder if for different $t_1$ and $t_2$, $X(t_1)$ is independent of $X(t_2)$. If it is, following $P(X_{t_1},\cdots,X_{t_k})=P(X_{t_1})\cdots P(X_{t_k})$, I can conclude that the joint distribution is actually a K-dimensional standard normal distribution.
- I've tried to show that $X(t_1)$ is independent of $X(t_2)$ by definition $P(X_{t_1},X_{t_2}) = P(X_{t_1})P(X_{t_2}\vert X_{t_1})$, but that yields a tricky integration.
Any help is appreciated.