Consider the sequences of mean zero random variables $(X_n)_{n \in \mathbb N}$ with finite variances, i.e., $E[X_n]=0$ and $E\left[X_n^2\right] =: \sigma_n< \infty$, for each $n$. Moreover, consider the $\ell^2$ sequence space with two norms: $||\cdot ||_{\infty}$ and $||\cdot ||_{2}$.
I am looking for an example of a sequence of random variables $(X_n)_{n \in \mathbb N}$ with variances $\sigma_n$'s as mentioned above and a sequence $(y_n)_{n \in \mathbb N}$ in $(\ell^2 , ||\cdot ||_{2})$ (i.e, $y_n =(y_{jn})_{j=0}^{\infty}$ with $\sum_{j=0}^\infty y_{jn}^2< \infty$, for each $n$ ) satisfying the following conditions:
- For each $n$, $i < j$ implies $|y_{jn}| < |y_{in}| $ (this is not very restrictive since $y_{jn}^2 \to 0$, as $j \to \infty$)
- Definig $x_n := \sigma_n^{1/2} y_n$, for all $n$, we have
$$ \lim_{n \to \infty}||x_n ||_{\infty} = 0 $$ - $(x_n)_{n \in \mathbb N}$ is a sequence such that: $$\lim_{n \to \infty}|| x_n ||_{2} = c, \quad c>0$$.
- Defining $p_{jn}:= \mathbb P \left[ |X_n| \leq 1/y_{jn} \right]$, we have: $$s_n := \sum_{j=0}^n p_{jn} \longrightarrow b < \infty,\quad (n \to \infty)\label{I}\tag{I}$$
Is it possible to find an example or is this not true?
Attempt
I tried adapting this example, but it doesn't work well.
Set $\sigma_n = \frac{1}{n+1}$ and $y_{jn}= \left(\frac{n}{n+1}\right)^{j}$. In this case, we have $$x_n = \sigma_n^{1/2} y_n = \sqrt{\frac{1}{n + 1}\left(\frac{n}{n + 1}\right)^{j}} $$ with $||x_n||_{2}=1\longrightarrow 1$ and $||x_n||_{\infty} = \sqrt{\frac{1}{n+1}} \longrightarrow 0$, as $n \to \infty$. Now it remains to choose suitable $X_n$ with variance $\sigma_n = \frac{1}{n+1}$ satisfying (\ref{I}). But I have the impression that this example is not very good, perhaps we need to find another example.