  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.
