independent increments of a stochastic process $N_t$ is a Poisson process, $X_t= \psi (-1)^{N_t}$, where $N_t$ and $\psi$ are independent,$P(\psi = 1)=P(\psi = -1)=1/2$,i want to prove that $X_t$ is a process with independent increments. My attemp: $X_{t_2}-X_{t_1}=\psi((-1)^{N_{t_2}}-(-1)^{N_{t_1}})=\psi(-1)^{N_{t_1}}((-1)^{N_{t_2}-N_{t_1}}-1)$, so i want to use the fact that
$N_{t_2}-N_{t_1}$ and $N_{t_1}$ are independent to prove that $X_{t_2}-X_{t_1}$ and $X_{t_1}$,but i dont know how to do it.
 A: The definition of a Poisson Process $\{N(t)\colon t > 0\}$ has several clauses, the first of which is that $N(t)$ is the number of arrivals in the time interval $(0,t]$ (note the delimiters very carefully), or, more wordily, the number of arrivals that occur after time $t =0$ and up to and including time $t$, and the name Poisson process is used because $N(t)$ is a Poisson random variable with parameter $\lambda t$ where $\lambda$ is called the arrival rate. More generally, for $t_1 < t_{2}$, $N(t_{2})-N(t_1)$ is the number of arrivals in $(t_1, t_{2}]$ and is a Poisson random variable with parameter $\lambda(t_{2}-t_1)$.
Another clause in the definition of a Poisson process is that
the increments (in the count of arrivals) that occur in disjoint intervals of time -- such as $(t_0,t_1],\quad (t_1, t_2],\quad \cdots \quad (t_{n-1},t_n]$
are mutually independent random variables. More wordily, the numbers of arrivals in these disjoint intervals of time are independent Poisson random variable with parameters $\lambda(t_{k+1}-t_k), k = 0, 1, 2, \ldots, n-1$.
Turning to $X(t) = \psi\cdot (-1)^{N(t)}$, note that $(-1)^{N(t)}$ alternates between $+1$ and $-1$, changing value at each Poisson arrival time $t_\gamma$, when $N(t_\gamma)$, the parity of the number of arrivals in $(0,t_\gamma]$ flips from odd to even or from even to odd. Since $\psi$ is equally likely to be $\pm 1$, $X(t) = \psi\cdot (-1)^{N(t)}$ merely ensures that the starting value $X(0)$ is random (we have what is called a random telegraph wave) instead the starting value always being $0$ corresponding to no arrivals thus far (which is called a semi-random telegraph wave).  See Probability, Random Variables, and Stochastic Processes by Papoulis and Pillai for more details.
Can you take it from here?
