Let $W$ be a random variable on $\mathbb{N}^{*}$ that follows the probability distribution $(\ell_k)_{k \in \mathbb{N}^*}$, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$. We admit that for all $B \subseteq \mathbb{N}^*$, $\mathbb{P}(W \in B) = \lim_{n \to \infty} \mathbb{P}(W_n \in B)$, and that if $X$ and $Y$ are two random variables taking values in $\mathbb{N}^*$ with $\mathbb{P}(X \in a\mathbb{N}^*) = \mathbb{P}(Y \in a\mathbb{N}^*)$ for all $a \in \mathbb{N}^*$, then $X$ and $Y$ have the same distribution. Specify the distribution of $W$. By considering $\ell_{1}$, what can we then conclude?
Let $W$ be a random variable on $\mathbb{N}^{*}$ that follows the probability distribution $(\ell_k)_{k \in \mathbb{N}^*}$, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
We admit that for all $B \subseteq \mathbb{N}^*$, $\mathbb{P}(W \in B) = \lim_{n \to \infty} \mathbb{P}(W_n \in B)$, and that if $X$ and $Y$ are two random variables taking values in $\mathbb{N}^*$ with $\mathbb{P}(X \in a\mathbb{N}^*) = \mathbb{P}(Y \in a\mathbb{N}^*)$ for all $a \in \mathbb{N}^*$, then $X$ and $Y$ have the same distribution.
Specify the distribution of $W$. By considering $\ell_{1}$, what can we then conclude?