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On Khinchin's theorem about the special role of the Gaussian distribution

Linda A. Khachatryan

Abstract

The purpose of this note is to recall one remarkable theorem of Khinchin about the special role of the Gaussian distribution. This theorem allows us to give a new interpretation of the Lindeberg condition: it guarantees the uniform integrability of the squares of normed sums of random variables and, thus, the passage to the limit under the expectation sign. The latter provides a simple proof of the central limit theorem for independent random variables.

On Khinchin's theorem about the special role of the Gaussian distribution

Abstract

The purpose of this note is to recall one remarkable theorem of Khinchin about the special role of the Gaussian distribution. This theorem allows us to give a new interpretation of the Lindeberg condition: it guarantees the uniform integrability of the squares of normed sums of random variables and, thus, the passage to the limit under the expectation sign. The latter provides a simple proof of the central limit theorem for independent random variables.
Paper Structure (7 theorems, 17 equations)

This paper contains 7 theorems, 17 equations.

Key Result

Theorem 1

Let $\{\xi_{n,j}\}$ be a double sequence of independent in each row random variables. If a limiting non-degenerate distribution for the sums $S_n$ exists, then for it to be Gaussian, it is necessary and sufficient that for any $\varepsilon>0$, random variables $\{\xi_{n,j}\}$ satisfy infsmall2.

Theorems & Definitions (12)

  • Theorem 1: Khinchin
  • Proposition 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 2
  • proof
  • Theorem 4: Lévy-Lindeberg
  • proof
  • ...and 2 more