**Properties**

- Since
*F*(*a*) = Pr(*X*≤*a*), the convergence in distribution means that the probability for*X*to be in a given range is approximately equal to the probability that the value of_{n}*X*is in that range, provided*n*is sufficiently large. - In general, convergence in distribution does not imply that the sequence of corresponding probability density functions will also converge. As an example one may consider random variables with densities
*ƒ*(_{n}*x*) = (1 − cos(2*πnx*))**1**_{{x∈(0,1)}}. These random variables converge in distribution to a uniform*U*(0, 1), whereas their densities do not converge at all. **Portmanteau lemma**provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. The lemma states that {*X*_{n}} converges in distribution to*X*if and only if any of the following statements are true:- Eƒ(
*X*) → Eƒ(_{n}*X*) for all bounded, continuous functions ƒ; - Eƒ(
*X*) → Eƒ(_{n}*X*) for all bounded, Lipschitz functions ƒ; - limsup{ Eƒ(
*X*) } ≤ Eƒ(_{n}*X*) for every upper semi-continuous function ƒ bounded from above; - liminf{ Eƒ(
*X*) } ≥ Eƒ(_{n}*X*) for every lower semi-continuous function ƒ bounded from below; - limsup{ Pr(
*X*∈_{n}*C*) } ≤ Pr(*X*∈*C*) for all closed sets*C*; - liminf{ Pr(
*X*∈_{n}*U*) } ≥ Pr(*X*∈*U*) for all open sets*U*; - lim{ Pr(
*X*∈_{n}*A*) } = Pr(*X*∈*A*) for all continuity sets*A*of random variable*X*.

- Eƒ(
**Continuous mapping theorem**states that for a continuous function*g*(·), if the sequence {*X*_{n}} converges in distribution to*X*, then so does {*g*(*X*_{n})} converge in distribution to*g*(*X*).**Lévy’s continuity theorem:**the sequence {*X*_{n}} converges in distribution to*X*if and only if the sequence of corresponding characteristic functions {*φ*} converges pointwise to the characteristic function_{n}*φ*of*X*.- Convergence in distribution is metrizable by the Lévy–Prokhorov metric.
- A natural link to convergence in distribution is the Skorokhod's representation theorem.

Read more about this topic: Convergence Of Random Variables, Convergence in Distribution

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