Linear Transformations in the Space $L^2$

Let $\xi =\left(\xi_{t}, t \in [0,1]\right)$ be a random process such that all $\xi_{t}, t \in T$, are jointly independent, identically distributed, and nontrivial (not equal to constants). Is the process $\xi$ stochastically continuous at even a single point of $[0,1]$?

Give an example of a process $\left(X_{t}, t \in [0,1]\right)$ that is stochastically continuous on $[0,1]$, but not integrable in probability on $[0,1]$ (the integral in probability is defined as convergence in probability of the Riemann integral sums).

Is the Poisson process $\left(N_{t}, t \geqslant 0\right)$ differentiable

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Find the distribution of the random variable $X_{t}=\int_{0}^{t} W_{s} d s$. Prove that the process $\left(X_{t}, t \geqslant 0\right)$ is Gaussian.

Find, for every $t \geq 0$, the joint distribution of the quantities $W_{t}$ and $X_{t}$ from the previous problem.

We are given the random process $X_{t}=\int_{0}^{t} e^{-W_{s}} d s$, where $W_{s}$ is a Wiener process. Find the mean function and covariance function of the process $X_{t}$.

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process. Compute, for $t>0$, the limit in $L^{2}$ as $n \rightarrow \infty$ of the expression