Theory of Statistics - George Mason University

Notes and Further Reading 143
After existence of a solution, the next question is whether the solution is
unique, or in our formulation of the problem, whether the moments uniquely
determine the probability distribution.
Many of the results concerning the moment problem involve probability
distributions that are not widely used. Heyde (1963) was the first to show
that a particular interesting distribution, namely the lognormal distribution,
was not uniquely determined by its moments. (This is Exercise 1.28.)
Corollary 1.18.1 is due to Thomas Stieltjes who proved it without use of
Theorem 1.18. Proofs and further discussion of the theorem and corollary can
be found in Shohat and Tamarkin (1943).
Sequences and Limit Theorems
The various forms of the central limit theorem have a long history of both
both the theory and the applications. Petrov (1995) provides an extensive
coverage. Dudley (1999) discusses many of the intricacies of the theorems and
gives extensions of the theory.
The most important seminal result on the limiting distributions of extreme
values was obtained by Fisher and Tippett (1928). von Mises (de Misès) (1939)
and Gnedenko (1943) cleaned up some of the details, and Theorem 1.59 is essentially
in the form stated in Gnedenko (1943). The limiting distributions of
extreme values are discussed at some length by David and Nagaraja (2003),
de Haan and Ferreira (2006), and Galambos (1978); and as mentioned in the
text, a proof of Theorem 1.59, though not the same as given by Gnedenko is
given by de Haan and Ferreira.
De Finetti’s theorem allows the extension of certain results for independent
sequences to similar results for exchangeable sequences. Taylor et al. (1985)
prove a number of limit theorems for sums of exchangeable random variables.
A quote from the Preface of Gnedenko and Kolmogorov (1954) is appropriate:
In the formal construction of a course in the theory of probability,
limit theorems appear as a kind of superstructure over elementary
chapters, in which all problems have finite purely arithmetical character.
In reality, however, the epistemological value of the theory of
probability is revealed only by limit theorems. Moreover, without limit
theorems it is impossible to understand the real content of the primary
concept of all our sciences — the concept of probability. In fact, all
epistemologic value of the theory of probability is based on this: that
large-scale random phenomena in their collective action create strict,
nonrandom regularity. The very concept of mathematical probability
would be fruitless if it did not find its realization in the frequency of
occurrence of events under large-scale repetition of uniform conditions
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Theory of Statistics c○2000–2013 James E. Gentle