1948

A Class of Statistics with Asymptotically Normal Distribution

Wassily Hoeffding

Annals of Mathematical Statistics, Vol. 19, No. 3. (Sep., 1948), pp. 293-325.

A Gap Theorem

M. Kac; R. Salem; A. Zygmund

Transactions of the American Mathematical Society, Vol. 63, No. 2. (Mar., 1948), pp. 235-243.

A Class of Statistics with Asymptotically Normal Distribution

Wassily Hoeffding

Annals of Mathematical Statistics, Vol. 19, No. 3. (Sep., 1948), pp. 293-325.

This paper gives useful conditions for easy checking that a statistic has an asymptotically normal distribution.The class of statistics (U-statistics) includes certain sums of multivariate functions of random variables. The proof substitutes the sum with a sum of independent random variables that could be in a certain sense considered as first order functional approximations to the multivariate functions. The sum of approximations goes to a Gaussian limit because of central limit theorem and the approximation converges to zero faster than the speed of convergence in the central limit theorem.

A Gap Theorem

M. Kac; R. Salem; A. Zygmund

Transactions of the American Mathematical Society, Vol. 63, No. 2. (Mar., 1948), pp. 235-243.

This is a paper about convergence properties of series of quasi-orthogonal functions. If the operator formed by covariances of a sequence of functions is bounded then the conditions for a.s. convergence are the same as for the series of orthogonal functions. The results give a nice way to prove central limit theorems for the second order stationary processes.