Hall, P.; Heyde, C. C.
Martingale limit theory and its application.
Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. xii+308 pp. ISBN: 0-12-319350-8
60-02 (60B12 60F05 60G42)
In retrospect, it is surprising that not until around 1970 was it realised
that the crucial condition for the central limit theorem (CLT) for
martingales $S_n=\sum_1^nX_i$ was $V_n^2/E(V_n^2)\rightarrow 1$, where
$V_n^2$ denotes the conditional variance $\sum_1^nE(X_i^2|\scr F_{i-1})$.
During the 1970s many extensions and refinements of this idea appeared in
the literature, and the need arose for a definitive reference work on
"weak" martingale limit theorems, as a complement to J. Neveu's book
[ Discrete-parameter martingales, English translation, North-Holland,
Amsterdam, 1975;
MR0402915 (53 \#6729)]
which treats only "strong" limit
theorems. This book addresses that need, in the discrete-time setting.
The book starts with a brisk treatment of the a.s. convergence theory, laws
of large numbers and square function inequalities. Chapter 3, on the CLT,
is the heart of the book. McLeish's slick characteristic function proof of
the basic result is used, and there follows discussion of convergence to
mixtures of normals, convergence of moments, rates of convergence and
reverse martingales. In Chapter 4 the Skorokhod embedding technique is used
to treat weak convergence of martingales to Brownian motion, and rates of
convergence are again given. Next the functional form of the law of the
iterated logarithm (LIL) is discussed. In Chapter 5 the authors discuss
limit theorems for stationary processes. Here Gordin's technique of
approximating stationary sequences by martingales is developed to establish
a CLT and functional CLT and LIL; also Ibragimov's CLT under the hypothesis
of vanishing maximal correlation is given. In Chapter 6 the authors
describe applications in statistics, specifically to the estimation of
parameters in stochastic processes. Martingales arise naturally in studying
the asymptotic properties of the MLE, but the book also gives applications
in conditional least squares estimation, and in estimating autocorrelations
in time series. Chapter 7, aptly entitled "Miscellaneous applications",
includes de Finetti's theorem for exchangeable sequences, Kingman's
subadditive ergodic theorem, stochastic approximation methods and several
other topics.
The reviewer liked the book very much. It gives a good view of the "state
of the art" in the theory, and a wide sample of applications; it should
succeed in becoming the standard reference work in this area. The
reviewer's only criticisms are of omissions, mainly of the lack of
discussion of the continuous-time case. Here it is known that essentially
the same results hold, with the role of the conditional variance $V_n^2$
being played by $\langle M,M\rangle_t$. A chapter on this would have made
the book substantially more useful as a reference work. Also, for
stationary sequences there are strong approximation techniques which give
stronger conclusions under stronger hypotheses: more discussion of the
range of applicability of the various hypotheses used on stationary
sequences would have been helpful. Finally, work by I. S. Helland [Z.
Wahrsch. Verw. Gebiete 52 (1980), no. 3, 251--265;
MR0576886 (81j:60045)]
on
weak convergence to Brownian motion appeared just too late to be noted in
the bibliography, which seems otherwise complete as far as the main "weak
limit theorem" topics are concerned.
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