(a) State the thinning and superposition properties of a Poisson process on $\mathbb{R}_{+}$. Prove the superposition property.

(b) A bi-infinite Poisson process $\left(N_{t}: t \in \mathbb{R}\right)$ with $N_{0}=0$ is a process with independent and stationary increments over $\mathbb{R}$. Moreover, for all $-\infty<s \leqslant t<\infty$, the increment $N_{t}-N_{s}$ has the Poisson distribution with parameter $\lambda(t-s)$. Prove that such a process exists.

(c) Let $N$ be a bi-infinite Poisson process on $\mathbb{R}$ of intensity $\lambda$. We identify it with the set of points $\left(S_{n}\right)$ of discontinuity of $N$, i.e., $N[s, t]=\sum_{n} \mathbf{l}\left(S_{n} \in[s, t]\right)$. Show that if we shift all the points of $N$ by the same constant $c$, then the resulting process is also a Poisson process of intensity $\lambda$.

Now suppose we shift every point of $N$ by $+1$ or $-1$ with equal probability. Show that the final collection of points is still a Poisson process of intensity $\lambda$. [You may assume the thinning and superposition properties for the bi-infinite Poisson process.]