Fourier Analysis in Probability Theory

673 Pages·1972·22.948 MB·English

by  Tatsuo Kawata, Z. W. Birnbaum and E. Lukacs (Auth.)

Probability and Mathematical Statistics
A Series of Monographs and Textbooks
Editors Z. W. Birnbaum E. Lukacs
University of Washington Bowling Green State University
Seattle, Washington Bowling Green, Ohio
1. Thomas Ferguson. Mathematical Statistics: A Decision Theoretic Approach. 1967
2. Howard Tucker. A Graduate Course in Probability. 1967
3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967
4. P. Révész. The Laws of Large Numbers. 1968
5. H. P. McKean, Jr. Stochastic Integrals. 1969
6. B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev. Mathematical Methods of
Reliability Theory. 1969
7. Demetrios A. Kappos. Probability Algebras and Stochastic Spaces. 1969
8. Ivan N. Pesin. Classical and Modern Integration Theories. 1970
9. S. Vajda. Probabilistic Programming. 1972
0. Sheldon M. Ross. Introduction to Probability Models. 1972
1. Robert B. Ash. Real Analysis and Probability. 1972
2. V. V. Fedorov. Theory of Optimal Experiments. 1972
3. K. V. Mardia. Statistics of Directional Data. 1972
14. H. Dym and H. P. McKean. Fourier Series and Integrals. 1972
15. Tatsuo Kawata. Fourier Analysis in Probability Theory. 1972
16. Fritz Oberhettinger. Fourier Transforms of Distributions and Their Inverses: A
Collection of Tables. 1973
17. Paul Erdös and Joel Spencer. Probabilistic Methods in Combinatorics. 1973
18. K. Sarkadi and I. Vincze. Mathematical Methods of Statistical Quality Control. 1973
19. Michael R. Anderberg. Cluster Analysis for Applications. 1973
20. W. Hengartner and R. Theodorescu. Concentration Functions. 1973
21. Kai Lai Chung. A Course in Probability Theory, Second Edition. 1974
22. L. H. Koopmans. The Spectral Analysis of Time Series. 1974
23. L. E. Maistrov. Probability Theory: A Historical Sketch. 1974
24. William F. Stout. Almost Sure Convergence. 1974
25. E. J. McShane. Stochastic Calculus and Stochastic Models. 1974
In Preparation
Z. Govindarajulu. Sequential Statistical Procedures
Roger Cuppens. Decomposition of Multivariate Probabilities
FOURIER ANAL YSIS
IN PROBABILITY THEORY
TATSUO KAWATA
Department of Engineering Science
Faculty of Engineering
Keio University, Yokohama, Japan
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Preface
The methods and results of Fourier analysis have been effectively
utilized in the analytic theory of probability. Moreover, simple analogs
of some results in Fourier analysis have actually given rise to many
significant results in probability theory. However, one often hears the
complaint that in seeking pertinent results from Fourier analysis which
are needed in the study of probability, the standard texts give a presenta-
tion that is, in most cases, too detailed to be useful. The authors primary
purpose, therefore, was to present useful results from the theories of
Fourier series, Fourier transforms, Laplace transforms, and other related
topics, in a fashion that will enable the student easily to find the results
and proofs he desires before he proceeds to more detailed investigations.
To further this purpose, particular attention has been given to clarifica-
tion of the interactions and analogies among these theories.
Chapters 1-8, present the elements of classical Fourier analysis, in
the context of their applications to probability theory. This is done in a
comprehensive but elementary fashion.
Chapters 9-14 are devoted to basic results from the theory of char-
acteristic functions of probability distributors, the convergence of distri-
bution functions in terms of characteristic functions, and series of
xi
Xll PREFACE
independent random variables. It is hoped that this presentation will
help the reader better to understand the workings of Fourier analysis.
It should also serve to exhibit some detailed classical results from these
fields.
The use of Fourier analysis has spread into almost all parts of proba-
bility theory, and it would be impossible to discuss all of these areas
within the scope of this book. For this reason, certain limitations have
been imposed. Some important and recent results such as the almost
everywhere convergence of the Fourier series of a function of Lp(—n π),
y
p > 1 could not be included ; even Lv and Hv theories with 1 < p, p φ 2
were not given since it was felt that to be too ambitious in scope would
jeopardize the use of the book.
One particular omission, contained in the original plan of the book,
is especially regretted by the author. That is the strong analogy between
the theory of Fourier series with gaps, or more general series with prop-
erties similar to gap conditions, and the theory of series of independent
random variables.
I am very grateful to Professor E. Lukacs for having given me the
opportunity to write this book and for his unfailing encouragement and
stimulation during my nine years as a professor at the Catholic Uni-
versity of America. I should like to extend my appreciation also to Dr. G.
R. Andersen, Mr. N. Monsour, and Dr. B. McDonagh who read the
manuscript, helped me in many respects and corrected the mistakes.
My thanks are also given to Mrs. J. Lawrence for typing the most part
of the manuscript, Mrs. J. Schäfer for her help in many respects, and
the Catholic University of America and the National Science Foundation
(GP-6175) for financial support.
I
Introduction
In this chapter we summarize the basic notions and results from real
and complex function theories and probability theory, that will be used
in this book.
1.1. Measurable Space; Probability Space
Let Ω be a set of elements (space) and let J^ be a class of subsets of Ω.
Disregarding the trivial case, we assume that Ω is nonempty. Suppose
that ssf contains the empty set 0 and is closed with respect to comple-
ments and with respect to countable unions; that is,
(i) if E e J^ then the complement Ec of E is a set of s/\
y
(ii) if {E n = 1, 2,... } is a countable sequence of sets of J^, then
ni
oo
Then J^ is called a or-field.
1
2 I. INTRODUCTION
J^ contains the space Ω.
Let C be any class of sets of Ω. Then there is the smallest or-field s/
which contains C, that is, for any cr-field M with M z> C , J / CM holds,
and oQ^is called the minimal σ-field over C. If S is the class of all subsets
of Ω> then S is a σ-field containing C and J^ is given by the intersection
of all σ-fields containing C.
Consider a fixed cr-field *$/. A couple (ß, s/) is called a measurable
space. Any set of ß which belongs to s/ is called a measurable set.
A nonnegative set function μ(2?), defined for all sets of J^ that satisfy
μ(0Ε ) = ξμ(Ε ) (1)
η η
\ n-1 / n-1
for any sequence 2^ of disjoint measurable sets, is called a measure on
a measurable space (Ω, μ)\ μ(Ε) could be + co. We suppose, disregarding
the extreme case, that for some £ e j /, μ(Ε) < oo. The property (1) is
also referred to as the countable additivity of a set function μ(Ε). From
(1) we may derive the basic properties of the measure:
(i) ,,(0) = 0. (2)
(ii) If E Ee Jtf, and E <=. E , then
lt z i 2
μ(Ε ) £ μ(Ε ), (3)
1 2
μ(Ε,)-μ(Ε ) = μ(Ε,-Ε ). (4)
1 1
(iii) If {E , n — 1, 2,... } is a sequence of sets of J^ (not necessarily
n
disjoint), then
μ(ΌΕ )^ξμ(Ε ). (5)
η η
\ n-1 / n-1
(iv) Suppose that
lim E = E, £ eJ/, »=1,2,...,
n n
n-too
that is,
E = lim sup E = lim inf E ,
n n
where
limsupi: = Π U£*>
n
η-κ» n—1 k—n
HmmfE = (J C\E .
n k
n-+Go n—1 λ—n
1.1. MEASURABLE SPACE; PROBABILITY SPACE 3
Then E 6 s/ and
\ΐτημ(Ε ) = μ(Ε). (6)
η
W->oo
A triplet (Ω> J^ μ) is a measure space (i2, J^) together with a measure
y
μ defined on J^.
If μ(ί2) < oo, then μ is called a finite measure. In this case, μ(2?) < oo
for all E e J/. If there is a sequence E ,n = 1,2,..., in J^ such that
n
oo
Ü=\JE„,
71 = 1
and μ(£" ) < oo, n = 1, 2,..., then μ is called a o-finite measure.
η
In probability theory, we deal with a measure space (ß, J/, P), where
P is a measure with Ρ{Ω) = 1. This triplet is called the probability space
and the measure P is called the probability or the probability measure.
Let St be a class, containing the empty set 0, which is closed with
respect to complements and with respect to finite unions, that is, E c e &
with E e& and for any finite sequence {E , n = 1, 2,..., m) of sets
n
of^,
m
\jE e^.
n
Then^ is called a field.
A measure m(E) on a fields is, by definition, a nonnegative set func-
tion defined on &t such that
(i) /n(0) = O,
(ii) if {E > n = 1, 2,... } is a sequence of disjoint sets of St such
n
that
then
where m(E) may be +oo. A finite or σ-finite measure o n^ is similarly
defined as in the case of measure on a σ-field.
We have the following extension theorem.
Theorem 1.1.1. Let m be a measure on a field St in Ω. Then there
exists a measure μ defined on the minimal σ-field s/ over St such that
μ(Ε) = m(E) for any E e SS.
4 I. INTRODUCTION
μ is finite or σ-finite according as m is finite or σ-finite. Such extension
is unique if m is σ-finite. (See, for example, Royden [1], p. 219.)
A measure space (Ω, s/ μ) is called complete if s/ contains all subsets
y
of sets whose measure is zero; that is, if E es/ana μ(Ε) = 0, then any
F c E is contained in s/. Of course μ(Ρ) = 0. In this case, μ is called
a complete measure.
Theorem 1.1.2. Let (Q,s/, μ) be any measure space. Then there al-
ways exists a complete measure space (ß, s/ , μ ) such that
0 0
(i) S/ŒS/0I
(ii) for any Ees/, μ(Ε) = μ (Ε).
0
Actually, s/ is defined from s/ by adding to s/ all the sets of the
0
form E u F, where E e s/ F is any subset of a set of s/ of measure
y
zero, and μ is defined to be μ (Ε u F) = μ(Ε), μ (Ε) = μ(Ε) for any
0 0 0
E e s/ and any F a subset of a set of s/ of measure zero.
Now we shall discuss the special case in which Ω is the Euclidean
space. A set of points x = (x x . . ., x )in the w-dimensional space R ,
l9 29 n n
such that a < x < b, i = 1, 2,..., n> where a b i = 1,2,...,« are
{ { t iy iy
given numbers, is called an open interval in R . The unique minimal
n
σ-field over the class consisting of all open intervals is called the Borel
σ-field or simply the Borel field. Sets of the Borel field are called Borel
sets or Borel measurable sets.
Consider the class C of all sets obtained by the finite number of set
operations (complements, unions, intersections) on open intervals in R .
n
Obviously, C is a field (not a σ-field). It is also obvious that the minimal
(T-field over C is identical with the Borel field ^.
Let F(x , x ,.. ., x ) be a finite-valued function defined over R . The
x 2 n n
nth difference of F is defined by
Ahnn Zfc1 - · - AH^F{X1,. . ., Xn_x , Xn) = JF(Ä?! +Αχ ,. . ., Xn-i + K-! , Χη + Κ)
f \ xl y χ2~\η2 > · · · » χη\ηη) * ' '
— F\XJ[-h,. . ., # -ι + /*η-1 » χη)
1 1 η
+ '·· + ···
+ (-Ι)^^,^,...,^); (8)
that is,
Διι r [χ ,. . ., χ ) = r [χ ,. . ., Xi-i, Χ{ -\- ni, Χι+ι,. . ., x )
1 η λ n
-Γ yXi,. . ., ΧΪ ,. . ., # J ,
η
4" zfc1 · · · Δ\Ψ = ΔΪΤ(Δ&( ■ ■ ■ Δ\Ψ) ···).
1.1. MEASURABLE SPACE; PROBABILITY SPACE 5
Suppose that
(i) F(x , x,..., x ) is nondecreasing for each x , i = 1, 2,. . ., n;
1 2 n {
(ii) Ah« Afc?· - .AÏF(x .. .,x ) ^ 0 for h .. .h ^ 0.
l9 n ly y n
Now define the set function Φ(Ε) on C in such a way that
Φ(Ε) = F{a + h - 0,..., a ^ + h _ -0,a + h - 0)
x x n n x n n
- F{a + 0, a + h - 0,.. ., a + h - 0)
x 2 2 n n
- F(a + K — 0,..., a _ + K-i — 0,a + 0)
1 n x n
... +(_l)»F(ii + 0,fl + 0,...,e + 0) (9)
+ 1 a JI
for intervals E: a < x < a + A, i = 1, 2,.. ., n and extend Φ(Ε) for
{ { t t y
all sets of C to be additive in a natural way. We then easily show that Φ(Ε)
so defined on C is a measure on the field C. Then, in view of Theorem
1.1.1, we may define a measure Φ on the Borel fields.
Φ is called a 5or^/ measure. Since F is finite-valued for any point in
R the measure Φ is σ-finite; and, if
ny
lim F(x x ..., x ), lim F ^, x,..., x ) (10)
ly 2y n 2 n
are finite and hence F(x ,. .., x ) is bounded, the measure Φ is a finite
x w
measure.
Let (R ^ Φ ) be the completion of (R ^ Φ) (the complete
ny y 0 ny y
measure space). The measure Φ is called the Lebesgue-Stielt]'es measure.
0
A set of i?7 is called a Lebesgue measurable set and i? the C/ÖW of Lebesgue
measurable sets.
When the completed measure Φ is specialized in such a way that
0
Φ{Ε) = ηη -. -h for E = {a < x < a + h i = 1, 2,.. ., n) it is
χ 2 n { i { iy y
called the Lebesgue measure on i? .
w
Throughout this book the Lebesgue measure of a set E will be denoted
by m(E) or mE.
If, besides, the conditions (i) and (ii) of F(x x . . ., x ) F(x x
ly 2y n y ly 2y
. . ., x ) satisfy the conditions that
n
(iii) lim __ F(x, x ,. . ., x ) = 0 for each z,
Xi ) 00 x 2 n
(iv) lim^.^_ ^+οο F{x, x,..., *„) = 1, and
f x 2
(v) ίχΛ^,. . ., x ) is left-continuous for each x ,
n {
then F{x x . . . x ) is called a distribution function.
x y 2y y n
Condition (v) is emphasized in probability theory. However in ana-
lytical considerations, (v) is sometimes ignored, since the measure Φ gen-