Let $\xi = (\xi_t,t ∈ [0,1])$ be a random process such that all $xi_t, t ∈ T$, are independent in the aggregate, equally distributed and non-trivial (different from constant). Is the process $xi$ stochastically continuous at a single point $[0,1]$? Stochastic continuity in point $t_0$ means that $\xi_t \overset{\mathbb{P}}{\to} \xi_{t_0}, t \rightarrow t_0$
Random variables are called independent in the aggregate if for any set of Borel sets $B_{t_1}, ..., B_{t_n} \in \mathbb{B(\mathbb{R})}$ $$ \mathbb{P}(\xi_{t_1} \in B_{t_1}, ..., \xi_{t_n} \in B_{t_n}) = \mathbb{P}(\xi_{t_1} \in B_{t_1}) ...\mathbb{P}(\xi_{t_n} \in B_{t_n}) $$
I would be grateful for any hint, since I don't understand how to even approach the problem.