let $N(t)$ be a poissonian counting process with parameter $\lambda$, we'll define $X(t)$ as a telegraph process in the following way: $$X(t) = B \cdot (-1)^{N_t}$$ where B gets values $\{-1,1\}$ with equal probability. $X(t)$ is statistically independent of $n(t)$. $n(t)$ is a guassian white noise with spectral power density $N_0$. We'll define $$v(t) = \int_0^t X(s) n(s) ds \quad t \geq 0$$
Find expectation and autocorrelation of $v(t)$, is $v(t)$ a guassian?
Where I'm struggling:
- the proof starts by defining $$\eta_t = \int_0^t g(s)n(s)ds$$ where $g(t)$ is a function that gets values $\pm 1$ and has a finite number of discontinuities on any finite interval (aka a sample function of a telegraph process) - what are we really doing here - why can't we just use $v(t)$ as is? when will I need to use this step? Can you please explain what $g(t)$ really is because i don't quite understand it?
2.The proof continues with computations I can follow and ends up with the following results: $$E(\eta_{t_1} \eta_{t_2}) = N_0 \text{min}(t_1,t_2)$$ $$E[\eta_t]=0$$ based on these results the solution asserts that the expectation and autocorrelation define completely the distribution law of $\eta_t$ and it independent of the transitions of $g(s)$ and their locations, and therefore $v(t)$ is a gaussian random process - I don't understand how they reached each conclusion here...