We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$ Show that $\left(J_{n}\right)_{n \in \mathbb{N}^{*}}$ is a well-defined sequence and that it is increasing and convergent. We will set $a_{k} = \frac{1}{2k-1}$ and express the expectation of $\left|T_{n}\right|$ using the method of question II.A.4.
We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let
$$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $\left(J_{n}\right)_{n \in \mathbb{N}^{*}}$ is a well-defined sequence and that it is increasing and convergent.
We will set $a_{k} = \frac{1}{2k-1}$ and express the expectation of $\left|T_{n}\right|$ using the method of question II.A.4.