grandes-ecoles 2022 Q28

grandes-ecoles · France · centrale-maths2__mp Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$), deduce that the event $$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$ also has probability 1. One may examine the series $\sum |S_{\phi(j+1)} - S_{\phi(j)}|$.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$), deduce that the event
$$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$
also has probability 1. One may examine the series $\sum |S_{\phi(j+1)} - S_{\phi(j)}|$.

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