Convergence in Distribution

Definition

• \(\{(\Omega_{n}, \mathcal{F}_{n}, P_{n})\}_{n \in \mathbb{N}}\),
  • probability space
• $(\Omega, \mathcal{F}, P)$,
  • probability space,
• $(S, \rho)$,
  • metric space
• \(\{X_{n}\}\),
  • r.v. on \((\Omega_{n}, \mathcal{F}_{n}, P_{n})\)
• $X$,
  • r.v. on \((\Omega, \mathcal{F}, P)\)

\(\{X_{n}\}_{n \in \mathbb{N}}\) is said to converges to $X$ in distribution and write $X_{n} \overset{d}{\rightarrow} X$ if \(\{P_{n}X_{n}^{-1}\}\) converges wealkly to $PX^{-1}$. That is, for all bounded continuous real-valued function $f$,

\[\lim_{n \rightarrow \infty} \mathrm{E}_{n} \left[ f(X_{n}) \right] = \mathrm{E} \left[ f(X) \right] .\]

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Reference