Lévy's continuity theorem

Lévy's continuity theorem   (Redirected from Lévy continuity theorem) Jump to navigation Jump to search In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem,[1] named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. This theorem is the basis for one approach to prove the central limit theorem and it is one of the major theorems concerning characteristic functions.

Statement Suppose we have a sequence of random variables {textstyle {X_{n}}_{n=1}^{infty }} , not necessarily sharing a common probability space, the sequence of corresponding characteristic functions {textstyle {varphi _{n}}_{n=1}^{infty }} , which by definition are {displaystyle varphi _{n}(t)=operatorname {E} left[e^{itX_{n}}right]quad forall tin mathbb {R} , forall nin mathbb {N} ,} where {displaystyle operatorname {E} } is the expected value operator.

If the sequence of characteristic functions converges pointwise to some function {displaystyle varphi } {displaystyle varphi _{n}(t)to varphi (t)quad forall tin mathbb {R} ,} then the following statements become equivalent: {displaystyle X_{n}} converges in distribution to some random variable X {displaystyle X_{n} {xrightarrow {mathcal {D}}} X,} i.e. the cumulative distribution functions corresponding to random variables converge at every continuity point of the c.d.f. of X; {textstyle {X_{n}}_{n=1}^{infty }} is tight: {displaystyle lim _{xto infty }left(sup _{n}operatorname {P} {big [},|X_{n}|>x,{big ]}right)=0;} {displaystyle varphi (t)} is a characteristic function of some random variable X; {displaystyle varphi (t)} is a continuous function of t; {displaystyle varphi (t)} is continuous at t = 0. Proof Rigorous proofs of this theorem are available.[1][2] References ^ Jump up to: a b Williams, D. (1991). Probability with Martingales. Cambridge University Press. section 18.1. ISBN 0-521-40605-6. ^ Fristedt, B. E.; Gray, L. F. (1996). A modern approach to probability theory. Boston: Birkhäuser. Theorems 14.15 and 18.21. ISBN 0-8176-3807-5. Categories: Probability theoremsTheorems in statisticsPaul Lévy (mathematician)