The ratio of moments in a normal distribution

Question

I'm reading a paper where they (Mann and Whitney) want to show the limiting distribution they get is normal. They do this by looking at a ratio of moments. They do a computation then conclude the limiting distribution is normal by a "well known theorem". Can someone provide a reference? The relevant part of the paper is copied below:

Answer 1

This fact is in Billingsley's Probability and Measure, although that's not how Mann and Whitney knew it. Section 30, "The Method of Moments", notes that the normal distribution is "determined by its moments", that is, is the only probability distribution with the same moments, and states Theorem 30.2, on page 344 in the first (1979) edition, and page 390 of the third (1995) edition:

Suppose that the distribution of $X$ is determined by its moments, that the $X_n$ have moments of all orders, and that $\lim_n E[X^r_n] = E[X^r]$ for $r=1,2,\ldots.$ Then $X_n\Rightarrow X$.

(Mann and Whitney possibly could have known the result as stated in Appendix II (see especially p.384) of Uspensky's 1937 Introduction to Mathematical Probability, which presents the Chebyshev theory of the method of moments described in a wikipedia article. This article is largely by Michael Hardy, who supplied the other answer to this question.)

M&W's business about "ratio of moments" is a notational paper tiger, an artifact of standardization. To show that $Y_n/\sigma(Y_n)=X_n$ converges in distribution to $X$ this way, Mann and Whitney verify (in separate even $r$ and odd $r$ cases) $E[Y_n^r]/E[Y_n^2]^{r/2}\to EX^r$, and so on.

The treatment of the limiting normality of the Mann-Whitney test is much slicker (= less ham-fisted) in Hájek and Šidák's Theory of Rank Tests.

Answer 2

I will surmise that $\operatorname E_{nm}(u^r)$ is an expected value of $u^r$ where $u$ is a random variable whose probability distribution is determined by parameters $n,m.$

It appears that all three of the following were relied upon:

• For a normal distribution with expectation $0$ and variance $\sigma^2$ the $r$th moment, when $r$ is even, is $(2r-1)(2r-3) \cdots 3 \cdot 5 \cdot 1 \cdot \sigma^{2r}.$
• If a probability distribution has the same moments as a normal distribution, then it is a normal distribution. (This fails to be true of some distributions. If I recall correctly, there are distributions that are not lognormal that have the same moments as a lognormal distribution.)
• If the sequence of moments of a distribution depending on $n,m$ converges pointwise to the sequence of moments of a normal distribution as $n,m\to\infty,$ then the distribution depending on $n,m$ converges in distribution to a normal distribution as $n,m\to\infty.$

I don't have references at hand at this moment, but maybe it's worthwhile to point out that any of the three might be called a well known theorem. Maybe from the context one could figure out which one they had in mind.

The first one can be proved by computing an integral, relying on $\Gamma(\tfrac 1 2)= \sqrt\pi$ plus elementary methods.

PS: Maybe just the first and third bullet points above are enough.