---- > [!definition] Definition. ([[convergence in distribution of random variables]]) > > - Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[probability|probability space]]. > - Let $E$ be a [[topological space]] (likely $\mathbb{R}$, $\mathbb{R}^{d}$, [[Polish space|Polish]], etc.). > - Let $\mathcal{P}(E)$ denote the collection of [[probability|probability measures]] on the state space $\big( E, \mathcal{B}(E) \big)$. > > A [[sequence]] of [[random variable|random variables]] $X_{1},X_{2},\dots:\Omega \to E$ is said to **converge in distribution** to $X: \Omega \to E$ if their [[probability distribution|distributions]] $\mathbb{P}_{X_{1}}, \mathbb{P}_{X_{2}},\dots$ weakly converge to $\mathbb{P}_{X}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```