Exercises and Solutions Manual for Integration and Probability: by Paul Malliavin
Autor Gerard Letac Traducere de L. Kayen Limba Engleză Paperback – 12 iun 1995
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Specificații
ISBN-13: 9780387944210
ISBN-10: 0387944214
Pagini: 142
Dimensiuni: 155 x 235 x 10 mm
Greutate: 0.27 kg
Ediția:1995
Editura: Springer
Colecția Springer
Locul publicării:New York, NY, United States
ISBN-10: 0387944214
Pagini: 142
Dimensiuni: 155 x 235 x 10 mm
Greutate: 0.27 kg
Ediția:1995
Editura: Springer
Colecția Springer
Locul publicării:New York, NY, United States
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Public țintă
ResearchCuprins
I
Measurable
Spaces
and
Integrable
Functions.-
1
?-algebras
and
partitions.-
2
r-families.-
3
Monotone
classes
and
independence.-
4
Banach
limits.-
5
A
strange
probability
measure.-
6
Integration
and
distribution
functions.-
7
Evaluating
$$
\sum\nolimits_{n
=
1}^\infty
{\frac{{{{\left(
{
-
1}
\right)}^n}}}{{{n^2}}}}
$$.-
8
Monotone
convergence.-
9
Vector
integration.-
10
Convergence
in
measure
and
composition
of
functions.-
11
Principle
of
separation
of
variables.-
12
The
Cauchy-Schwarz
inequality.-
13
Test
that
X
?
Y
almost
everywhere.-
14
Image
of
a
measure.-
15
Primitives
of
square
integrable
functions.-
II
Borel
Measures
and
Radon
Measures.-
1
Positive
measures
on
an
open
interval.-
2
Distribution
functions.-
3
Convexity
and
growth.-
4
Convexity
and
measure.-
5
Integral
representation
of
positive
convex
functions
(0,
?).-
6
Integral
representations
of
Askey
functions.-
7
Gauss’s
inequality.-
8
Integral
of
a
decreasing
function.-
9
Second
mean
value
theorem
for
integrals.-
10
Variance
of
a
distribution
on
[0,
1].-
11
Variance
of
the
distribution
of
a
convex
function
on
[0,
1].-
12
Rational
functions
which
preserve
Lebesgue
measure.-
13
A
measure
on
the
half-plane.-
14
Weak
convergence
and
moments.-
15
Improper
integrals
and
Lebesgue
measure.-
16
$$
\int_0^\infty
{\frac{{\sin
x}}{x}}
dx,\int_0^\infty
{\left(
{\cos
ax
-
\cos
bx}
\right)}
\frac{{dx}}{x},\int_0^\infty
{\left(
{\cos
ax
-
\cos
bx}
\right)}
\frac{{dx}}{{{x^2}}}
$$.-
17
Comparisons
between
different
Lp
spaces.-
18
Differentiation
under
the
integral
sign.-
19
Laplace
transform
of
a
measure
on
[0,+?).-
20
Comparison
of
vague,
weak,
and
narrow
convergence.-
21
Weak
compactness
of
measures.-
22
Vague
convergence
and
limit
of
µn
(0).-
23
Vague
convergence
and
restriction
to
a
closed
set.-
24
Change
of
variables
in
an
integral.-
25
Image
of
a
measure
and
the
Jacobian.-
III
Fourier
Analysis.-
1
Characterizations
of
radial
measures.-
2
Radial
measures
and
independence.-
3
Area
of
the
sphere.-
4
Fourier
transform
of
the
Poisson
kernel
of
R+n+1.-
5
Askey-Polya
functions.-
6
Symmetric
convex
sets
in
the
plane
and
measures
on
[0,?).-
7
T.
Ferguson’s
theorem.-
8
A
counterexample
of
Herz.-
9
Riesz
kernels.-
10
Measures
on
the
circle
and
holomorphic
functions.-
11
Harmonic
polynomials
and
the
Fourier
transform.-
12
Bernstein’s
inequality.-
13
Cauchy’s
functional
equation.-
14
Poisson’s
formula.-
15
A
list
of
Fourier-Plancheral
transforms.-
16
Fourier-Plancheral
transform
of
a
rational
function.-
17
Computing
some
Fourier-Plancheral
transforms.-
18
Expressing
the
Fourier-Plancheral
transform
as
a
limit.-
19
An
identity
for
the
Fourier-Plancheral
transform.-
20
The
Hilbert
transform
on
L2(R).-
21
Action
of
L1(R)
on
L2(R).-
22
Another
expression
for
the
Hilbert
transform.-
23
A
table
of
properties
of
the
Hilbert
transform.-
24
Computing
some
Hilbert
transforms.-
25
The
Hilbert
transform
and
distributions.-
26
Sobolev
spaces
on
R.-
27
H.
Weyl’s
inequality.-
IV
Hilbert
Space
Methods
and
Limit
Theorems
in
Probability
Theory.-
1
Fancy
dice.-
2
The
geometric
distribution.-
3
The
binomial
and
Poisson
distributions.-
4
Construction
of
given
distributions.-
5
Von
Neumann’s
method.-
6
The
laws
of
large
numbers.-
7
Etemadi’s
method.-
8
A
lemma
on
the
random
walks
Sn.-
9
?(s)
=
limn??
(P[Sn
?
s·n])1/n
exists.-
10
Evaluating
?(s)
in
some
concrete
cases.-
11
Algebra
of
the
gamma
and
beta
distributions.-
12
The
gamma
distribution
and
the
normal
distribution.-
13
The
Cauchy
distribution
and
the
normal
distribution.-
14
A
probabilistic
proof
of
Stirling’s
formula.-
15
Maxwell’s
theorem.-
16
If
X1
and
X2
are
independent,
then
$$\frac{{\left(
{{x_1},{x_2}}
\right)}}{{{{\left(
{x_1^2
+
x_2^2}
\right)}^{1/2}}}}$$
is
uniform.-
17
Isotropy
of
pairs
and
triplets
of
independent
variables.-
18
The
only
invertible
distributions
are
concentrated
at
a
point.-
19
Isotropic
multiples
of
normal
distributions.-
20
Poincaré’s
lemma.-
21
Schoenberg’s
theorem.-
22
A
property
of
radial
distributions.-
23
Brownian
motion
hits
a
hyperplane
in
a
Cauchy
distribution.-
24
Pittinger’s
inequality.-
25
Cylindrical
probabilities.-
26
Minlos’s
lemma.-
27
Condition
that
a
cylindrical
probability
be
a
probability
measure.-
28
Lindeberg’s
theorem.-
29
H.
Chernoff’s
inequality.-
30
Gebelein’s
inequality.-
31
Fourier
transform
of
the
Hermite
polynomials.-
32
Another
definition
of
conditional
expectation.-
33
Monotone
continuity
of
conditional
expectations.-
34
Concrete
computation
of
conditional
expectations.-
35
Conditional
expectations
and
independence.-
36
E(X|Y)
=
Y
and
E(Y|X)
=
X.-
37
Warnings
about
conditional
expectations.-
38
Conditional
expectations
in
the
absolutely
continuous
case
and
the
Gaussian
case.-
39
Examples
of
martingales.-
40
A
reversed
martingale.-
41
A
probabilistic
approximation
of
an
arithmetic
conjecture.-
42
A
criterion
for
uniform
integrability.-
43
The
Galton-Watson
process
and
martingales.-
V
Gaussian
Sobolev
Spaces
and
Stochastic
Calculus
of
Variations.-
1
d
and
?
cannot
both
be
continuous.-
2
Growth
of
the
Hermite
polynomials.-
3
Viskov’s
lemma.-
4
Cantelli’s
conjecture.-
5
Lancaster
probabilities
in
R2.-
6
Sarmanov’s
theorem.