Skorokhod's representation theorem

Skorokhod's representation theorem In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Soviet mathematician A. V. Skorokhod.

Statement Let {displaystyle (mu _{n})_{nin mathbb {N} }} be a sequence of probability measures on a metric space {displaystyle S} such that {displaystyle mu _{n}} converges weakly to some probability measure {displaystyle mu _{infty }} on {displaystyle S} as {displaystyle nto infty } . Suppose also that the support of {displaystyle mu _{infty }} is separable. Then there exist {displaystyle S} -valued random variables {displaystyle X_{n}} defined on a common probability space {displaystyle (Omega ,{mathcal {F}},mathbf {P} )} such that the law of {displaystyle X_{n}} is {displaystyle mu _{n}} for all {displaystyle n} (including {displaystyle n=infty } ) and such that {displaystyle (X_{n})_{nin mathbb {N} }} converges to {displaystyle X_{infty }} , {displaystyle mathbf {P} } -almost surely.

See also Convergence in distribution References Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc. ISBN 0-471-19745-9. (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem) Categories: Probability theoremsTheorems in statistics