grandes-ecoles 2022 Q18

grandes-ecoles · France · x-ens-maths-b__mp Discrete Random Variables Properties of Probability Measures and Convergence of Measures
Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:
  1. [i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
  2. [ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} P(r \mid X_n) = P(r \mid X)$.
Show that then the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$. 
Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:
\begin{enumerate}
\item[i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
\item[ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} P(r \mid X_n) = P(r \mid X)$.
\end{enumerate}
Show that then the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$.
Paper Questions
Q1a Q1b Q1c Q1d Q2a Q2b Q3a Q3b Q4 Q5a Q5b Q6 Q7a Q7b Q7c Q8a Q8b Q9 Q10a Q10b Q10c Q11 Q12a Q12b Q12c Q12d Q12e Q13 Q14 Q15a Q15b Q15c Q16 Q17a Q17b Q17c Q18 Q19 Q20a Q20b Q21