Basic Concepts of the Theory of Stochastic Processes

Let $X = X(\omega , t)$ be a function defined on $\Omega \times T$ taking values in $\mathbb {R}$. Let, moreover, $E$ be the set of all functions $f: T \rightarrow \mathbb {R}$. Consider on this set the cylindrical $\sigma$-algebra $\mathcal{E} = \sigma \left(\left\{ f: f(t) \in B\right\} , B \in \mathscr {B}\left(\mathbb {R}\right), t \in T\right)$. Prove that $(X_{t} = X(\cdot , t), t \in T)$ is a random process if and only if $X$ is an $(\mathcal{F} \mid \mathcal{E})$-measurable function.

The processes $\left(X_{t}, t \in T\right)$ and $\left(Y_{t}, t \in \tilde{T}\right)$ are called independent if the $\sigma$-algebras they generate are independent. Prove that $\left(X_{t}, t \in T\right)$ and $\left(Y_{t}, t \in \tilde{T}\right)$ are independent if and only if, for any natural numbers $n, m$ and for any $t_{1}, \ldots , t_{n} \in T, \tilde{t}_{1}, \ldots , \tilde{t}_{m} \in \tilde{T}$, the vectors $\left(X_{t_{1}}, \ldots , X_{t_{n}}\right)$ and $\left(Y_{\tilde{t}_{1}}, \ldots , Y_{\tilde{t}_{m}}\right)$ are independent.

Draw a typical trajectory of the process $Y_{t}$ in the Sparre Andersen insurance model.

Let $\left(X_{t}, t \geqslant 0\right)$ be a renewal process, with $\mathbb {P}\left(\xi_k < +\infty \right) = 1 \; \forall k \in \mathbb {N}$. Prove that

Let $\left(X_{t}, t \geqslant 0\right)$ be a renewal process built from independent, but not necessarily identically distributed, random variables $\left(\xi_{n}, n \in \mathbb {Z}\right)$ taking values in $[0, +\infty ]$.

Prove the CLT for the renewal process.

Let $(\xi_n, n \in \mathbb {N})$ be a random sequence whose elements $\xi_1, \xi_2, \ldots$ are independent random variables. Denote by

the tail $\sigma$-algebra, i.e., the intersection of all $\sigma$-algebras generated by the collections $\left\{ \xi_k, k \geqslant n+1\right\}$. Prove the zero-one law: for any event $A \in \mathfrak {X}$, either $\mathbb {P}\left(A\right) = 0$ or $\mathbb {P}\left(A\right) = 1$.