# Convergence of Random Variables - Convergence in Distribution - Properties

Properties

• Since F(a) = Pr(Xa), the convergence in distribution means that the probability for Xn to be in a given range is approximately equal to the probability that the value of 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 ƒn(x) = (1 − cos(2πnx))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 {Xn} converges in distribution to X if and only if any of the following statements are true:
• Eƒ(Xn) → Eƒ(X) for all bounded, continuous functions ƒ;
• Eƒ(Xn) → Eƒ(X) for all bounded, Lipschitz functions ƒ;
• limsup{ Eƒ(Xn) } ≤ Eƒ(X) for every upper semi-continuous function ƒ bounded from above;
• liminf{ Eƒ(Xn) } ≥ Eƒ(X) for every lower semi-continuous function ƒ bounded from below;
• limsup{ Pr(XnC) } ≤ Pr(XC) for all closed sets C;
• liminf{ Pr(XnU) } ≥ Pr(XU) for all open sets U;
• lim{ Pr(XnA) } = Pr(XA) for all continuity sets A of random variable X.
• Continuous mapping theorem states that for a continuous function g(·), if the sequence {Xn} converges in distribution to X, then so does {g(Xn)} converge in distribution to g(X).
• Lévy’s continuity theorem: the sequence {Xn} converges in distribution to X if and only if the sequence of corresponding characteristic functions {φn} converges pointwise to the characteristic function 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.

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