Probability with martingales:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2008
|
| Edition: | 11. print. |
| Series: | Cambridge mathematical textbooks
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 243 - 245. - Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XV, 251 S. |
| ISBN: | 0521406056 9780521406055 |
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MARC
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| 100 | 1 | |a Williams, David |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Probability with martingales |c David Williams |
| 250 | |a 11. print. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2008 | |
| 300 | |a XV, 251 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge mathematical textbooks | |
| 500 | |a Literaturverz. S. 243 - 245. - Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 0 | 7 | |a Wahrscheinlichkeitstheorie |0 (DE-588)4079013-7 |2 gnd |9 rswk-swf |
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| _version_ | 1804138700866584576 |
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| adam_text | Contents
Preface
—
please read!
xi
A Question of Terminology
xiii
A Guide to Notation
xiv
Chapter
0:
A Branching-Process Example
1
0.0.
Introductory remarks.
0.1.
Typical number of children, X.
0.2.
Size
of nth generation, Zn.
0.3.
Use of conditional expectations.
0.4.
Extinction
probability,
тт.
0.5.
Pause for thought: measure.
0.6.
Our first martingale.
0.7.
Convergence (or not) of expectations.
0.8.
Finding the distribution of
Moo·
0.9.
Concrete example.
PART A: FOUNDATIONS
Chapter
1:
Measure Spaces
14
1.0.
Introductory remarks.
1.1.
Definitions of algebra, ^-algebra.
1.2.
Ex¬
amples.
Borei
σ
-algebras,
B(S),
В
=
B(R).
1.3.
Definitions concerning
set functions.
1.4.
Definition of measure space.
1.5.
Definitions con¬
cerning measures.
1.6.
Lemma. Uniqueness of extension, Tr-systems.
1.7.
Theorem. Caratheodory s extension theorem.
1.8.
Lebesgue measure
Leb
on
((0,1], 0(0,1]). 1.9.
Lemma. Elementary inequalities.
1.10.
Lemma.
Monotone-convergence properties of measures.
1.11.
Example/Warning.
Chapter
2:
Events
23
2.1.
Model for experiment:
(Ω,^,Ρ).
2.2.
The intuitive meaning.
2.3-
Examples of
(Ω,
ƒ*)
pairs.
2.4.
Almost surely (a.s.)
2.5.
Reminder:
lim
sup,
lim inf
,
J.
lim,
etc.
2.6.
Definitions,
lim
sup En,
(Εη,ί.ο.).
2.7.
vi
Contents
First Borei-Cantelli Lemma
(ВСІ).
2.8.
Definitions,
liminf
E„,(E„,ev).
2.9.
Exercise.
Chapter
3:
Random Variables
29
3.1.
Definitions.
Σ
-measurable
function,
πιΣ,
(mS)+,bE.
3.2.
Elementary
Propositions on measurability.
3.3.
Lemma. Sums and products of mea¬
surable functions are measurable.
3.4.
Composition Lemma.
3.5.
Lemma
on measurability of infs,
lim inf
s
of functions.
3.6.
Definition. Random
variable.
3.7.
Example. Coin tossing.
3.8.
Definition,
σ
-algebra
generated
by a collection of functions on
Ω.
3.9.
Definitions. Law, Distribution Func¬
tion.
3.10.
Properties of distribution functions.
3.11.
Existence of random
variable with given distribution function.
3.12.
Skorokod
representation of
a random variable with prescribed distribution function.
3.13.
Generated
σ
-algebras
- a discussion.
3.14.
The Monotone-Class Theorem.
Chapter
4:
Independence
38
4.1.
Definitions of independence.
4.2.
The jr-system Lemma; and the
more familiar definitions.
4.3.
Second
Borei-Cantelli
Lemma (BC2).
4.4.
Example.
4.5.
A fundamental question for modelling.
4.6.
A coin-tossing
model with applications.
4.7.
Notation: IID RVs.
4.8.
Stochastic processes;
Markov chains.
4.9.
Monkey typing Shakespeare.
4.10.
Definition. Tail
σ-
algebras.
4.11.
Theorem. Kolmogorov s
0-1
law.
4.12.
Exercise/Warning.
Chapter
5:
Integration
49
5.0.
Notation, etc.
μ(/)
:=: ƒ
/άμ, μ(/;Α).
5.1.
Integrals of non-negative
simple functions, SF+.
5.2.
Definition of
//(ƒ), ƒ
Є
(mE)+.
5.3.
Monotone-
Convergence Theorem
(MON).
5.4.
The Fatou Lemmas for functions (FA-
TOU).
5.5.
Linearity .
5.6.
Positive and negative parts of
ƒ. 5.7.
Inte¬
grable
function, ^(S.S,
μ).
5.8.
Linearity.
5.9.
Dominated Convergence
Theorem (DOM).
5.10.
Scheffé s
Lemma
(SCHEFFÉ).
5.11.
Remark on
uniform integrability.
5.12.
The standard machine.
5.13.
Integrals over
subsets.
5.14.
The measure
f
μ,
ƒ €
(mE)+.
Chapter
6:
Expectation
58
Introductory remarks.
6.1.
Definition of expectation.
6.2.
Convergence
theorems.
6.3.
The notation E(Jr;F).
6.4.
Markov s inequality.
6.5.
Sums of non-negative RVs.
6.6.
Jensen s inequality for convex functions.
6.7.
Monotonicity
of
&
norms.
6.8.
The
Schwarz
inequality.
6.9.
C2:
Pythagoras, covariance, etc.
6.10.
Completeness of
С
(1 <
ρ
<
oo).
6.11.
Orthogonal projection.
6.12.
The elementary formula for expectation.
6.13.
Holder from Jensen.
Contenti vii
Chapter
7:
An Easy Strong Law
71
7.1.
Independence means multiply
-
again!
7.2.
Strong Law
-
first version.
7.3.
Chebyshev s inequality.
7.4.
Weierstrass
approximation theorem.
Chapter
8:
Product Measure
75
8.0.
Introduction and advice.
8.1.
Product measurable structure,
Σι χ Σ2.
8.2.
Product measure, Pubini s Theorem.
8.3.
Joint laws, joint pdfs.
8.4.
Independence and product measure.
8.5.
S(R)
= ß(Rn). 8.6.
The n-fold
extension.
8.7.
Infinite products of probability triples.
8.8.
Technical note
on the existence of joint laws.
PART B: MARTINGALE THEORY
Chapter
9:
Conditional Expectation
83
9.1.
A motivating example.
9.2.
Fundamental Theorem and Definition
(Kolmogorov,
1933). 9.3.
The intuitive meaning.
9.4.
Conditional ex¬
pectation as least-squares-best predictor.
9.5.
Proof of Theorem
9.2. 9.6.
Agreement with traditional expression.
9.7.
Properties of conditional ex¬
pectation: a list.
9.8.
Proofs of the properties in Section
9.7. 9.9.
Regular
conditional probabilities and pdfs.
9.10.
Conditioning under independence
assumptions.
9.11.
Use of symmetry: an example.
Chapter
10:
Martingales
93
10.1.
Filtered spaces.
10.2.
Adapted processes.
10.3.
Martingale, super-
martingale,
submartingale.
10.4.
Some examples of martingales.
10.5.
Fair
and unfair games.
10.6.
Previsible
process, gambling strategy.
10.7.
A fun¬
damental principle: you can t beat the system!
10.8.
Stopping time.
10.9.
Stopped
supermartingales
are
supermartingales.
10.10.
Doob s Optional-
Stopping Theorem.
10.11.
Awaiting the almost inevitable.
10.12.
Hitting
times for simple random walk.
10.13.
Non-negative superharmonic func¬
tions for Markov chains.
Chapter
11:
The Convergence Theorem
106
11.1.
The picture that says it all.
11.2.
Upcrossings.
11.3.
Doob s Upcross-
ing Lemma.
11.4.
Corollary.
11.5.
Doob s Forward Convergence Theorem.
11.6.
Warning.
11.7.
Corollary.
vüi
Contents
Chapter
12:
Martingales bounded in £2
110
12.0.
Introduction.
12.1.
Martingales in C2: orthogonality of increments.
12.2.
Sums of zero-mean independent random variables in C2.
12.3.
Ran¬
dom signs.
12.4.
A symmetrization technique: expanding the sample space.
12.5.
Kolmogorov s Three-Series Theorem.
12.6.
Cesàro s
Lemma.
12.7.
Kronecker s Lemma.
12.8.
A Strong Law under variance constraints.
12.9.
Kolmogorov s Truncation Lemma.
12.10.
Kolmogorov s Strong Law of
Large Numbers (SLLN).
12.11.
Doob decomposition.
12.12.
The angle-
brackets process (M).
12.13.
Relating convergence of
M
to finiteness of
(M)«,.
12.14.
A trivial Strong Law for martingales
inc2.
12.15.
Levy s
extension of the
Borei-Cantelli
Lemmas.
12.16.
Comments.
Chapter
13:
Uniform Integrability
126
13.1.
An absolute continuity property.
13.2.
Definition. UI family.
13.3.
Two simple sufficient conditions for the UI property.
13.4.
UI property
of conditional expectations.
13.5.
Convergence in probability.
13.6.
Ele¬
mentary proof of (BDD).
13.7.
A necessary and sufficient condition for C1
convergence.
Chapter
14:
UI Martingales
133
14.0.
Introduction.
14.1.
UI martingales.
14.2.
Levy s Upward Theorem.
14.3.
Martingale proof of Kolmogorov s
0-1
law.
14.4.
Levy s Downward
Theorem.
14.5.
Martingale proof of the Strong Law.
14.6.
Doob s Sub-
martingale Inequality.
14.7.
Law of the Iterated Logarithm: special case.
14.8.
A standard estimate on the normal distribution.
14.9.
Remarks on ex¬
ponential bounds; large deviation theory.
14.10.
A consequence of Holder s
inequality.
14.11.
Doob s C* inequality.
14.12.
Kakutani s Theorem on
product martingales. 14.13.The
Radon-Nikodým
theorem.
14.14.
The
Radon-Nikodým
theorem and conditional expectation.
14.15.
Likelihood
ratio; equivalent measures.
14.16.
Likelihood ratio and conditional expec¬
tation.
14.17.
Kakutani s Theorem revisited; consistency of LR test.
14.18.
Note on Hardy spaces, etc.
Chapter
15:
Applications
153
15.0.
Introduction
-
please read!
15.1.
A trivial martingale-representation
result.
15.2.
Option pricing; discrete Black-Scholes formula.
15.3.
The
Mabinogion sheep problem.
15.4.
Proof of Lemma
15.3(c).
15.5.
Proof
of result 15.3(d).
15.6.
Recursive nature of conditional probabilities.
15.7.
Bayes
formula for
bi
variate
normal distributions.
15.8.
Noisy observation of
a single random variable.
15.9.
The Kalman-Bucy filter.
15.10.
Harnesses
entangled.
15.11.
Harnesses unravelled,
1. 15.12.
Harnesses unravelled,
2.
Contents
ix
PART
С:
CHARACTERISTIC FUNCTIONS
Chapter
16:
Basic Properties of CFs
172
16.1.
Definition.
16.2.
Elementary properties.
16.3.
Some uses of char¬
acteristic functions.
16.4.
Three key results.
16.5.
Atoms.
16.6.
Levy s
Inversion Formula.
16.7.
A table.
Chapter
17:
Weak Convergence
179
17.1.
The elegant definition.
17.2.
A practical formulation.
17-3.
Sko-
rokhod representation.
17.4.
Sequential compactness for Prob(R).
17.5.
Tightness.
Chapter
18:
The Central Limit Theorem
185
18.1.
Levy s Convergence Theorem.
18.2.
о
and
О
notation.
18.3.
Some
important estimates.
18.4.
The Central Limit Theorem.
18.5.
Example.
18.6.
CF proof of Lemma
12.4.
APPENDICES
Chapter
Al:
Appendix to Chapter
1 192
Al.l. A non-measurable subset A of S1. A1.2. d-systems. A1.3. Dynkin s
Lemma. Al.4. Proof of Uniqueness Lemma
1.6.
A1.5.
λ
-sets:
algebra
case. A1.6. Outer measures.
Al.
7.
Carathéodory s
Lemma. A1.8. Proof of
Carathéodory s
Theorem.
Al
.9.
Proof of the existence of Lebesgue measure
on
((0,
1},B(Q,
1]).
Al.
10.
Example of non-uniqueness of extension. Al.ll.
Completion of a measure space.
Al.
12.
The Baire category theorem.
Chapter
A3:
Appendix to Chapter
3 205
A3.1. Proof of the Monotone-Class Theorem
3.14.
A3.2. Discussion of
generated er-algebras.
Chapter
A4:
Appendix to Chapter
4 208
A4.1. Kolmogorov s Law of the Iterated Logarithm. A4.2. Strassen s Law
of the Iterated Logarithm. A4.3. A model
for a
Markov chain.
Chapter A5: Appendix to Chapter
5 211
АбЛ.
Doubly monotone arrays.
Â5.2.
The key use of Lemma 1.10(a).
A5.3. Uniqueness of integral . A5.4. Proof of the Monotone-Convergence
Theorem.
x
Contents
Chapter A9: Appendix to Chapter
9 214
A9.1. Infinite products: setting things up. A9.2. Proof of A9.1(e).
Chapter A13: Appendix to Chapter
13 217
A13.1. Modes of convergence: definitions. A13.2. Modes of convergence:
relationships.
Chapter A14: Appendix to Chapter
14 219
ARI.
The a-algebra-Fr,
Γ
a stopping time. A14.2. A special case of
OST.
A14.3. Doob s Optional-Sampling Theorem for
Ш
martingales. A14.4. The
result for UI
submartingales.
Chapter A16: Appendix to Chapter
16 222
A16.1. DiiFerentiation under the integral sign.
Chapter E: Exercises
224
References
243
Index
246
|
| any_adam_object | 1 |
| author | Williams, David |
| author_facet | Williams, David |
| author_role | aut |
| author_sort | Williams, David |
| author_variant | d w dw |
| building | Verbundindex |
| bvnumber | BV035372608 |
| classification_rvk | SK 800 SK 820 |
| ctrlnum | (OCoLC)612663522 (DE-599)BVBBV035372608 |
| discipline | Mathematik |
| edition | 11. print. |
| format | Book |
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| id | DE-604.BV035372608 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:32:22Z |
| institution | BVB |
| isbn | 0521406056 9780521406055 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017176963 |
| oclc_num | 612663522 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XV, 251 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge mathematical textbooks |
| spelling | Williams, David Verfasser aut Probability with martingales David Williams 11. print. Cambridge Cambridge Univ. Press 2008 XV, 251 S. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical textbooks Literaturverz. S. 243 - 245. - Hier auch später erschienene, unveränderte Nachdrucke Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Martingaltheorie (DE-588)4168982-3 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Martingal (DE-588)4126466-6 gnd rswk-swf Martingal (DE-588)4126466-6 s Martingaltheorie (DE-588)4168982-3 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 Wahrscheinlichkeitstheorie (DE-588)4079013-7 s 2\p DE-604 3\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017176963&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Williams, David Probability with martingales Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Martingaltheorie (DE-588)4168982-3 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Martingal (DE-588)4126466-6 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4168982-3 (DE-588)4064324-4 (DE-588)4126466-6 |
| title | Probability with martingales |
| title_auth | Probability with martingales |
| title_exact_search | Probability with martingales |
| title_full | Probability with martingales David Williams |
| title_fullStr | Probability with martingales David Williams |
| title_full_unstemmed | Probability with martingales David Williams |
| title_short | Probability with martingales |
| title_sort | probability with martingales |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Martingaltheorie (DE-588)4168982-3 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Martingal (DE-588)4126466-6 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Martingaltheorie Wahrscheinlichkeitsrechnung Martingal |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017176963&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT williamsdavid probabilitywithmartingales |