Holdings: Probability :: THWS Bibkatalog

Probability: a graduate course

Saved in:

Bibliographic Details
Main Author:	Gut, Allan 1944- (Author)
Format:	Book
Language:	English
Published:	New York, NY Springer 2007
Edition:	Corr. 2. print.
Series:	Springer texts in statistics
Subjects:	Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie Lehrbuch Lehrbuch - Wahrscheinlichkeitstheorie
Online Access:	Inhaltsverzeichnis
Physical Description:	XXIII, 603 S.
ISBN:	9780387228334 0387228330

Staff View

MARC


LEADER	00000nam a2200000 c 4500
001	BV023096699
003	DE-604
005	20111102
007	t
008	080124s2007 \|\|\|\| 00\|\|\| eng d
015			\|a 04,N32,0556 \|2 dnb
020			\|a 9780387228334 \|9 978-0-387-22833-4
020			\|a 0387228330 \|9 0-387-22833-0
035			\|a (OCoLC)254222247
035			\|a (DE-599)BVBBV023096699
040			\|a DE-604 \|b ger \|e rakddb
041	0		\|a eng
049			\|a DE-384 \|a DE-20 \|a DE-11 \|a DE-91G \|a DE-824
082	0		\|a 519.2
082	0		\|a 510
084			\|a SK 800 \|0 (DE-625)143256: \|2 rvk
084			\|a 510 \|2 sdnb
084			\|a MAT 600f \|2 stub
100	1		\|a Gut, Allan \|d 1944- \|e Verfasser \|0 (DE-588)110855329 \|4 aut
245	1	0	\|a Probability \|b a graduate course \|c Allan Gut
250			\|a Corr. 2. print.
264		1	\|a New York, NY \|b Springer \|c 2007
300			\|a XXIII, 603 S.
336			\|b txt \|2 rdacontent
337			\|b n \|2 rdamedia
338			\|b nc \|2 rdacarrier
490	0		\|a Springer texts in statistics
650	0	7	\|a Wahrscheinlichkeitsrechnung \|0 (DE-588)4064324-4 \|2 gnd \|9 rswk-swf
650	0	7	\|a Wahrscheinlichkeitstheorie \|0 (DE-588)4079013-7 \|2 gnd \|9 rswk-swf
655		7	\|0 (DE-588)4123623-3 \|a Lehrbuch \|2 gnd-content
655		4	\|a Lehrbuch - Wahrscheinlichkeitstheorie
689	0	0	\|a Wahrscheinlichkeitsrechnung \|0 (DE-588)4064324-4 \|D s
689	0		\|5 DE-604
689	1	0	\|a Wahrscheinlichkeitstheorie \|0 (DE-588)4079013-7 \|D s
689	1		\|5 DE-604
856	4	2	\|m Digitalisierung UB Augsburg \|q application/pdf \|u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016299494&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA \|3 Inhaltsverzeichnis
999			\|a oai:aleph.bib-bvb.de:BVB01-016299494

Record in the Search Index

_version_	1804137352579252224
adam_text	Contents Preface .................................................. V Outline of Contents .....................................XVII Notation and Symbols ..................................XXI Introductory Measure Theory ............................ 1 1 Probability Theory. An Introduction ...................... 1 2 Basics from Measure Theory ............................. 2 2.1 Sets .............................................. 3 2.2 Collections of Sets .................................. 5 2.3 Generators ........................................ 7 2.4 A Metatheorem and Some Consequences .............. 9 3 The Probability Space ................................... 10 3.1 Limits and Completeness ............................ 11 3.2 An Approximation Lemma .......................... 13 3.3 The Borei Sets on E ................................ 14 3.4 The Borei Sets on R ............................... 16 4 Independence; Conditional Probabilities ................... 16 4.1 The Law of Total Probability: Bayes Formula ......... 17 4.2 Independence of Collections of Events ................. 18 4.3 Pair-wise Independence ............................. 19 •5 The Kolmogorov Zero-one Law ........................... 20 6 Problems .............................................. 22 Random Variables ....................................... 25 1 Definition and Basic Properties ........................... 25 1.1 Functions of Random Variables ...................... 28 2 Distributions ........................................... 30 2.1 Distribution Functions .............................. 30 2.2 . Integration: A Preview .............................. 32 X Cont (mts 2.3 Decomposition of Distributions ....................... 36 2.4 Some Standard Discrete Distributions ................. 39 2.5 Some Standard Absolutely Continuous Distributions .... 40 ■ 2. G The Cantor Distribution ............................ 40 2.7 Two Perverse Examples ............................. 42 3 Random Vectors: Random Elements ....................... 43 3.1 Random Vectors ................................... 43 3.2 Random Elements .................................. 45 4 Expectation: Definitions and Basics ....................... 46 4.1 Definitions ........................................ 46 .4.2 Basic Properties .................................... 48 5 Expectation; Convergence ................................ 54 6 Indefinite Expectations .................................. 58 7 A Change of Variables Formula ........................... 60 8 Moments, Mean, Variance ................................ 62 9 Product Spaces: Fubini s Theorem ........................ 64 9.1 Finite-dimensional Product Measures ................. 64 9.2 Fubini s Theorem .................................. 65 9.3 Partial Integration .................................. 66 9.4 The Convolution Formula ........................... G 7 10 Independence .......................................... 68 10.1 Independence of Functions of Random Variables ........ 71 10.2 Independence of σ- Algebras ......................... 71 10.3 Pair-wise Independence ............................. 71 10.4 The Kolmogorov Zero-one Law Revisited .............. 72 11 The Cantor Distribution ................................. 73 12 Tail Probabilities and Moments ........................... 74 13 Conditional Distributions ................................ 79 14 Distributions with Random Parameters .................... 81 15 Sums of a Random Number of Random Variables ........... 83 15.1 Applications ....................................... 85 16 Random Walks; Renewal Theory .......................... 88 •16.1 Random Walks ..................................... 88 16.2 Renewal Theory .................................... 89 16.3 Renewal Theory for Random Walks ................... 90 16.4 The Likelihood Ratio Test ........................... 91 16.5 Sequential Analysis ................................. 91 16.6 Replacement Based on Age .......................... 92 17 Extremes: Records ...................................... 93 17.1 Extremes .......................................... 93 17.2 Records ........................................... 93 18 Borei-Cantelli Lemmas .................................. 96 18.1 The Borel-Cantclli Lemmas 1 and 2 .................. 96 18.2 Some (Very) Elementary Examples ................... 98 18.3 Records ........................................... 101 ι οι it ei its XI 18.4 Recurrence and Transience of Simple Random Walks . . . 102 18.5 Eľ=i P(An) = °° and P (An i.o.) = 0.................104 18.6 Pair-wise Independence .............................104 18.7 Generalizations Without Independence; ................105 18.8 Extremes ..........................................107 18.9 Further Generalizations .............................109 19 A Convolution Table ....................................113 20 Problems ..............................................114 Inequalities ..............................................119 1 Tail Probabilities Estimated via Moments ..................119 2 Moment Inequalities ....................................127 3 Covariance; Correlation ..................................130 4 Interlude on ¿ -spaces ...................................131 5 Convexity ..............................................132 G Symmetrizatiori .........................................133 7 Probability Inequalities for Maxima .......................138 8 The Marcinkicwics-Zygmund Inequalities ..................146 9 Rosenthaľs Inequality ...................................151 10 Problems ..............................................153 Characteristic Functions .................................157 1 Definition and Basics ....................................157 1.1 Uniqueness; Inversion ...............................159 1.2 Multiplication .....................................164 1.3 Some Further Results ...............................165 2 Some Special Examples ..................................166 2.1 The Cantor Distribution ............................166 2.2 The Convolution Table Revisited .....................168 2.3 The Cauchy Distribution ............................170 2.4 Symmetric Stablo Distributions ......................171 2.5 Parsevaľs Relation .................................172 3 Two Surprises ..........................................173 4 Refinements ............................................175 5 Characteristic Functions of Random Vectors ................180 5.1. The Multivariate Normal Distribution ................180 5.2 The Mean and the Sample Variance Are Independent . . . 183 6 The Cumulant Generating Function .......................184 7 The Probability Generating Function ......................186 7.1 Random Vectors ...................................188 8 The Moment Generating Function ........................189 8.1 Random Vectors ...................................191 8.2 Two Boundary Cases ...............................191 9 Sums of a Random Number of Random Variables ...........192 10 The Moment Problem ...................................194 XII Contents ЮЛ The Moment Problem for Random Sums ..............196 11 Problems ..............................................197 5 Convergence .............................................201 1 Definitions .............................................202 1.1 Continuity Points and Continuity Sets ................203 1.2 Measurability ......................................205 1.3 Some Examples ....................................206 2 Uniqueness .............................................207 3 Relations Between Convergence Concepts ..................209 3.1 Converses .........................................212 4 Uniform Integrability ....................................214 5 Convergence of Moments ................................218 •5.1 Almost Sure Convergence ...........................218 5.2 Convergence in Probability ..........................220 5.3 Convergence in Distribution .........................222 6 Distributional Convergence Revisited ......................225 6.1 Scheffe s Lemma ...................................226 7 A Subsequence Principle .................................229 8 Vague Convergence: Helly s Theorem ......................230 8.1 Vague Convergence .................................231 8.2 Helly s Selection Principle ...........................232 8.3 Vague Convergence and Tightness ....................234 8.4 The Method of Moments ............................237 9 Continuity Theorems ....................................238 9.1 The Characteristic Function .........................238 9.2 The Cumulant Generating Function ..................240 9.3 The (Probability) Generating Function ................241 9.4 The Moment Generating Function ....................242 10 Convergence of Functions of Random Variables .............243 10.1 The Continuous Mapping Theorem ...................245 11 Convergence of Sums of Sequences ........................247 11.1 Applications .......................................249 11.2 Converses .........................................252 11.3 Symmetrization and Desy nimetrizat ion ................255 12 Cauchy Convergence ....................................256 13 Skorohod s Representation Theorem .......................258 14 Problems ..............................................260 6 The Law of Large Numbers ..............................265 1 Preliminaries ...........................................266 1.1 Convergence Equivalence ............................266 1.2 Distributional Equivalence ...........................267 1.3 Sums and Maxima .................................268 1.4 Moments and Tails .................................268 C oulent.s XIII 2 A Weak Law for Partial Maxima ...........................269 3 The Weak Law of Large Numbers .........................270 3.1 Two Applications .................................. 27G 4 A Weak Law Without Finito Mean ........................278 4.1 The St.. Petersburg Game ...........................283 5 Convergence of Series ...................................284 5.1 The Kolmogorov Convergence Criterion ...............286 5.2 A Preliminary Strong Law ...........................288 5.3 The Kolmogorov Three-series Theorem ................289 5.4 Levy s Theorem on the Convergence of Series ..........292 6 The Strong Law of Large Numbers ........................294 7 The Marcinkiewicz-Zygnmnd Strong Law ..................298 8 Randomly Indexed Sequences ............................301 9 Applications ...........................................305 9.1 Normal Numbers ...................................305 9.2 The Glivenko-CantelH Theorem ......................306 9.3 Renewal Theory for Random Walks ...................306 9.4 Records ...........................................307 10 Uniform Integrability: Moment Convergence ................309 11 Complete Convergence ..................................311 11.1 The Hsu-Robbhis-Erdös Strong Law ..................312 11.2 Complete Convergence and the Strong Law ............314 12 Some Additional Results and Remarks .....................315 12.1 Convergence Rates .................................315 12.2 Counting Variables .................................320 12.3 The Case r = p Revisited ............................321 12.4 Random Indices ....................................322 13 Problems ..............................................323 The Central Limit Theorem ..............................329 1 The i.i.d. Case ..........................................330 2 The Lindeberg-Lévy-Feller Theorem .......................330 2.1 Lyapounov s Condition ..............................339 2.2 Remarks and Complements ..........................340 2.3 Pair-wise Independence .............................343 2.4 The Central Limit Theorem for Arrays ................344 3 Anscombe s Theorem ....................................345 4 Applications ...........................................348 4.1 The Delta Method ..................................349 4.2 Stirling s Formula ..................................350 4.3 Renewal Theory for Random Walks ...................350 4.4 Records ...........................................351 5 Uniform Integrability: Moment Convergence ................352 6 Remainder Term Estimates ..............................354 6.1 The Berrv-Esseen Theorem ..........................355 XIV Contents 6.2 Proof of the Berry-Esseen Theorem 6.2................357 7 Some Additional Results and Remarks .....................362 7.1 Rates of Rates .....................................362 7.2 Non-uniform Estimates .............................363 7.3 Renewal Theory ....................................364 7.4 Records ...........................................364 7.5 Local Limit Theorems ..............................365 7.6 Large Deviations ...................................365 7.7 Convergence Rates .................................366 7.8 Precise Asymptotics ................................371 7.9 A Short Outlook on Extensions ......................374 8 Problems ..............................................376 8 The Law of the Iterated Logarithm ......................383 1 The Kolmogorov and Hartman-Wint nor LILs ...............384 .1.1 Outline of Proof ....................................385 2 Exponential Bounds .....................................385 3 Proof of the Hartman- Wintner Theorem ...................387 4 Proof of the Converse ...................................396 5 The LIL for Subsequences ................................398 5.1 A Borei-Cantelli Sum for Subsequences ...............401 5.2 Proof of Theorem 5.2...............................402 5.3 Examples .........................................404 6 Cluster Sets ............................................404 6.1 Proofs ............................................406 7 Some Additional Results and Remarks .....................412 7.1 Hartman- Wintner via Berry-Esseen ...................412 7.2 Examples Not Covered by Theorems 5.2 and 5.1 .......413 7.3 Further Remarks on Sparse Subsequences .............414 7.4 An Anscombe LIL ..................................416 7.5 Renewal Theory for Random Walks ...................417 7.6 Record Times ......................................417 7.7 Convergence Rates .................................418 7.8 Precise Asymptotics ................................419 7.9 The Other LIL .....................................419 8 Problems ..............................................420 9 Limit Theorems; Extensions and Generalizations .........423 1 Stable Distributions .....................................424 2 The Convergence to Types Theorem ......................427 3 Domains of Attraction ...................................430 3.1 Sketch of Preliminary Steps ..........................433 3.2 Proof of Theorems 3.2 and 3.3.......................435 3.3 Two Examples .....................................438 3.4 Two Variations ....................................439 <. outonts XV 3.5 Additional Results ..................................440 4 Infinitely Divisi lile Distributions ..........................442 5 S unis of Dependent Random Variables .....................448 6 Convergence of Extremes ................................451 6.1 Max-stable and Extremal Distributions ...............451 6.2 Domains of Attraction ..............................456 6.3 Record Values .....................................457 7 The Stein-Chen Method .................................459 8 Problems ..............................................464 10 Martingales ..............................................467 1 Conditional Expectation .................................468 1.1 Properties of Conditional Expectation ................471 1.2 Smoothing ........................................474 1.3 The Rao-Blackwell Theorem .........................475 1.4 Conditional Moment Inequalities .....................476 2 Martingale Definitions ...................................477 2.1 The Defining Relation ..............................479 2.2 Two Equivalent Definitions ..........................479 3 Examples ..............................................481 4 Orthogonality ..........................................487 5 Decompositions .........................................489 6 Stopping Times .........................................491 7 Doob s Optional Sampling Theorem .......................495 8 Joining and Stopping Martingales .........................497 9 Martingale Inequalities ..................................501 10 Convergence ...........................................508 10.1 Garsia s Proof .....................................508 10.2 The Upcrossings Proof ..............................511 10.3 Some Remarks on Additional Proofs ..................513 10.4 Some Questions ....................................514 10.5 A Non-convergent Martingale ........................515 10.6 A Central Limit Theorem? ..........................515 11 The Martingale {E(Z \| ƒ*„)} .............................515 12 Regular Martingales and Submartingales ...................516 12.1 A Main Martingale Theorem .........................517 12.2 A Main Submartingale Theorem .....................518 12.3 Two Non-regular Martingales ........................519 12.4 Regular Martingales Revisited .......................519 13 The Kolmogorov Zero-one Law ...........................520 14 Stopped Random Walks .................................521 14.1 Finiteness of Moments ..............................521 14.2 The Wald Equations ................................522 14.3 Tossing a Coin Until Success .........................525 14.4 The Gambler s Ruin Problem ........................526 XVI Contents 14.5 A Converse ........................................529 15 Regularity .............................................531 15.1 First Passage Times for Random Walks ...............535 15.2 Complements ......................................537 15.3 The Wald Fundamental Identity ......................538 16 Reversed Martingales and Submartingales ..................541 1G.1 The Law of Large1 Numbers ..........................544 16.2 fZ-statistics ........................................547 17 Problems ..............................................548 A Some Useful Mathematics ................................555 1 Taylor Expansion .......................................555 2 Mill s Ratio ............................................558 3 Sums and Integrals ......................................559 4 ■ Sums and Products .....................................560 5 Convexity; Clarkson s Inequality ..........................561 6 Convergence of (Weighted) Averages ......................564 7 Regularly and Slowly Varying Functions ...................566 8 Cauchy s Functional Equation ............................568 9 Functions and Dense Sets ................................570 References .....................................................577 Index ..........................................................589
adam_txt	Contents Preface . V Outline of Contents .XVII Notation and Symbols .XXI Introductory Measure Theory . 1 1 Probability Theory. An Introduction . 1 2 Basics from Measure Theory . 2 2.1 Sets . 3 2.2 Collections of Sets . 5 2.3 Generators . 7 2.4 A Metatheorem and Some Consequences . 9 3 The Probability Space . 10 3.1 Limits and Completeness . 11 3.2 An Approximation Lemma . 13 3.3 The Borei Sets on E . 14 3.4 The Borei Sets on R" . 16 4 Independence; Conditional Probabilities . 16 4.1 The Law of Total Probability: Bayes' Formula . 17 4.2 Independence of Collections of Events . 18 4.3 Pair-wise Independence . 19 •5 The Kolmogorov Zero-one Law . 20 6 Problems . 22 Random Variables . 25 1 Definition and Basic Properties . 25 1.1 Functions of Random Variables . 28 2 Distributions . 30 2.1 Distribution Functions . 30 2.2 . Integration: A Preview . 32 X Cont (mts 2.3 Decomposition of Distributions . 36 2.4 Some Standard Discrete Distributions . 39 2.5 Some Standard Absolutely Continuous Distributions . 40 ■ 2. G The Cantor Distribution . 40 2.7 Two Perverse Examples . 42 3 Random Vectors: Random Elements . 43 3.1 Random Vectors . 43 3.2 Random Elements . 45 4 Expectation: Definitions and Basics . 46 4.1 Definitions . 46 .4.2 Basic Properties . 48 5 Expectation; Convergence . 54 6 Indefinite Expectations . 58 7 A Change of Variables Formula . 60 8 Moments, Mean, Variance . 62 9 Product Spaces: Fubini's Theorem . 64 9.1 Finite-dimensional Product Measures . 64 9.2 Fubini's Theorem . 65 9.3 Partial Integration . 66 9.4 The Convolution Formula . G 7 10 Independence . 68 10.1 Independence of Functions of Random Variables . 71 10.2 Independence of σ- Algebras . 71 10.3 Pair-wise Independence . 71 10.4 The Kolmogorov Zero-one Law Revisited . 72 11 The Cantor Distribution . 73 12 Tail Probabilities and Moments . 74 13 Conditional Distributions . 79 14 Distributions with Random Parameters . 81 15 Sums of a Random Number of Random Variables . 83 15.1 Applications . 85 16 Random Walks; Renewal Theory . 88 •16.1 Random Walks . 88 16.2 Renewal Theory . 89 16.3 Renewal Theory for Random Walks . 90 16.4 The Likelihood Ratio Test . 91 16.5 Sequential Analysis . 91 16.6 Replacement Based on Age . 92 17 Extremes: Records . 93 17.1 Extremes . 93 17.2 Records . 93 18 Borei-Cantelli Lemmas . 96 18.1 The Borel-Cantclli Lemmas 1 and 2 . 96 18.2 Some (Very) Elementary Examples . 98 18.3 Records . 101 ι" 'οι it ei its XI 18.4 Recurrence and Transience of Simple Random Walks . . . 102 18.5 Eľ=i P(An) = °° and P (An i.o.) = 0.104 18.6 Pair-wise Independence .104 18.7 Generalizations Without Independence; .105 18.8 Extremes .107 18.9 Further Generalizations .109 19 A Convolution Table .113 20 Problems .114 Inequalities .119 1 Tail Probabilities Estimated via Moments .119 2 Moment Inequalities .127 3 Covariance; Correlation .130 4 Interlude on ¿''-spaces .131 5 Convexity .132 G Symmetrizatiori .133 7 Probability Inequalities for Maxima .138 8 The Marcinkicwics-Zygmund Inequalities .146 9 Rosenthaľs Inequality .151 10 Problems .153 Characteristic Functions .157 1 Definition and Basics .157 1.1 Uniqueness; Inversion .159 1.2 Multiplication .164 1.3 Some Further Results .165 2 Some Special Examples .166 2.1 The Cantor Distribution .166 2.2 The Convolution Table Revisited .168 2.3 The Cauchy Distribution .170 2.4 Symmetric Stablo Distributions .171 2.5 Parsevaľs Relation .172 3 Two Surprises .173 4 Refinements .175 5 Characteristic Functions of Random Vectors .180 5.1. The Multivariate Normal Distribution .180 5.2 The Mean and the Sample Variance Are Independent . . . 183 6 The Cumulant Generating Function .184 7 The Probability Generating Function .186 7.1 Random Vectors .188 8 The Moment Generating Function .189 8.1 Random Vectors .191 8.2 Two Boundary Cases .191 9 Sums of a Random Number of Random Variables .192 10 The Moment Problem .194 XII Contents ЮЛ The Moment Problem for Random Sums .196 11 Problems .197 5 Convergence .201 1 Definitions .202 1.1 Continuity Points and Continuity Sets .203 1.2 Measurability .205 1.3 Some Examples .206 2 Uniqueness .207 3 Relations Between Convergence Concepts .209 3.1 Converses .212 4 Uniform Integrability .214 5 Convergence of Moments .218 •5.1 Almost Sure Convergence .218 5.2 Convergence in Probability .220 5.3 Convergence in Distribution .222 6 Distributional Convergence Revisited .225 6.1 Scheffe's Lemma .226 7 A Subsequence Principle .229 8 Vague Convergence: Helly's Theorem .230 8.1 Vague Convergence .231 8.2 Helly's Selection Principle .232 8.3 Vague Convergence and Tightness .234 8.4 The Method of Moments .237 9 Continuity Theorems .238 9.1 The Characteristic Function .238 9.2 The Cumulant Generating Function .240 9.3 The (Probability) Generating Function .241 9.4 The Moment Generating Function .242 10 Convergence of Functions of Random Variables .243 10.1 The Continuous Mapping Theorem .245 11 Convergence of Sums of Sequences .247 11.1 Applications .249 11.2 Converses .252 11.3 Symmetrization and Desy nimetrizat ion .255 12 Cauchy Convergence .256 13 Skorohod's Representation Theorem .258 14 Problems .260 6 The Law of Large Numbers .265 1 Preliminaries .266 1.1 Convergence Equivalence .266 1.2 Distributional Equivalence .267 1.3 Sums and Maxima .268 1.4 Moments and Tails .268 C'oulent.s XIII 2 A Weak Law for Partial Maxima .269 3 The Weak Law of Large Numbers .270 3.1 Two Applications . 27G 4 A Weak Law Without Finito Mean .278 4.1 The St. Petersburg Game .283 5 Convergence of Series .284 5.1 The Kolmogorov Convergence Criterion .286 5.2 A Preliminary Strong Law .288 5.3 The Kolmogorov Three-series Theorem .289 5.4 Levy's Theorem on the Convergence of Series .292 6 The Strong Law of Large Numbers .294 7 The Marcinkiewicz-Zygnmnd Strong Law .298 8 Randomly Indexed Sequences .301 9 Applications .305 9.1 Normal Numbers .305 9.2 The Glivenko-CantelH Theorem .306 9.3 Renewal Theory for Random Walks .306 9.4 Records .307 10 Uniform Integrability: Moment Convergence .309 11 Complete Convergence .311 11.1 The Hsu-Robbhis-Erdös Strong Law .312 11.2 Complete Convergence and the Strong Law .314 12 Some Additional Results and Remarks .315 12.1 Convergence Rates .315 12.2 Counting Variables .320 12.3 The Case r = p Revisited .321 12.4 Random Indices .322 13 Problems .323 The Central Limit Theorem .329 1 The i.i.d. Case .330 2 The Lindeberg-Lévy-Feller Theorem .330 2.1 Lyapounov's Condition .339 2.2 Remarks and Complements .340 2.3 Pair-wise Independence .343 2.4 The Central Limit Theorem for Arrays .344 3 Anscombe's Theorem .345 4 Applications .348 4.1 The Delta Method .349 4.2 Stirling's Formula .350 4.3 Renewal Theory for Random Walks .350 4.4 Records .351 5 Uniform Integrability: Moment Convergence .352 6 Remainder Term Estimates .354 6.1 The Berrv-Esseen Theorem .355 XIV Contents 6.2 Proof of the Berry-Esseen Theorem 6.2.357 7 Some Additional Results and Remarks .362 7.1 Rates of Rates .362 7.2 Non-uniform Estimates .363 7.3 Renewal Theory .364 7.4 Records .364 7.5 Local Limit Theorems .365 7.6 Large Deviations .365 7.7 Convergence Rates .366 7.8 Precise Asymptotics .371 7.9 A Short Outlook on Extensions .374 8 Problems .376 8 The Law of the Iterated Logarithm .383 1 The Kolmogorov and Hartman-Wint nor LILs .384 .1.1 Outline of Proof .385 2 Exponential Bounds .385 3 Proof of the Hartman- Wintner Theorem .387 4 Proof of the Converse .396 5 The LIL for Subsequences .398 5.1 A Borei-Cantelli Sum for Subsequences .401 5.2 Proof of Theorem 5.2.402 5.3 Examples .404 6 ' Cluster Sets .404 6.1 Proofs .406 7 Some Additional Results and Remarks .412 7.1 Hartman- Wintner via Berry-Esseen .412 7.2 Examples Not Covered by Theorems 5.2 and 5.1 .413 7.3 Further Remarks on Sparse Subsequences .414 7.4 An Anscombe LIL .416 7.5 Renewal Theory for Random Walks .417 7.6 Record Times .417 7.7 Convergence Rates .418 7.8 Precise Asymptotics .419 7.9 The Other LIL .419 8 Problems .420 9 Limit Theorems; Extensions and Generalizations .423 1 Stable Distributions .424 2 The Convergence to Types Theorem .427 3 Domains of Attraction .430 3.1 Sketch of Preliminary Steps .433 3.2 Proof of Theorems 3.2 and 3.3.435 3.3 Two Examples .438 3.4 Two Variations .439 <. 'outonts XV 3.5 Additional Results .440 4 Infinitely Divisi lile Distributions .442 5 S unis of Dependent Random Variables .448 6 Convergence of Extremes .451 6.1' Max-stable and Extremal Distributions .451 6.2 Domains of Attraction .456 6.3 Record Values .457 7 The Stein-Chen Method .459 8 Problems .464 10 Martingales .467 1 Conditional Expectation .468 1.1 Properties of Conditional Expectation .471 1.2 Smoothing .474 1.3 The Rao-Blackwell Theorem .475 1.4 Conditional Moment Inequalities .476 2 Martingale Definitions .477 2.1 The Defining Relation .479 2.2 Two Equivalent Definitions .479 3 Examples .481 4 Orthogonality .487 5 Decompositions .489 6 Stopping Times .491 7 Doob's Optional Sampling Theorem .495 8 Joining and Stopping Martingales .497 9 Martingale Inequalities .501 10 Convergence .508 10.1 Garsia's Proof .508 10.2 The Upcrossings Proof .511 10.3 Some Remarks on Additional Proofs .513 10.4 Some Questions .514 10.5 A Non-convergent Martingale .515 10.6 A Central Limit Theorem? .515 11 The Martingale {E(Z \| ƒ*„)} .515 12 Regular Martingales and Submartingales .516 12.1 A Main Martingale Theorem .517 12.2 A Main Submartingale Theorem .518 12.3 Two Non-regular Martingales .519 12.4 Regular Martingales Revisited .519 13 The Kolmogorov Zero-one Law .520 14 Stopped Random Walks .521 14.1 Finiteness of Moments .521 14.2 The Wald Equations .522 14.3 Tossing a Coin Until Success .525 14.4 The Gambler's Ruin Problem .526 XVI Contents 14.5 A Converse .529 15 Regularity .531 15.1 First Passage Times for Random Walks .535 15.2 Complements .537 15.3 The Wald Fundamental Identity .538 16 Reversed Martingales and Submartingales .541 1G.1 The Law of Large1 Numbers .544 ' 16.2 fZ-statistics .547 17 Problems .548 A Some Useful Mathematics .555 1 Taylor Expansion .555 2 Mill's Ratio .558 3 Sums and Integrals .559 4 ■ Sums and Products .560 5 Convexity; Clarkson 's Inequality .561 6 Convergence of (Weighted) Averages .564 7 Regularly and Slowly Varying Functions .566 8 Cauchy's Functional Equation .568 9 Functions and Dense Sets .570 References .577 Index .589
any_adam_object	1
any_adam_object_boolean	1
author	Gut, Allan 1944-
author_GND	(DE-588)110855329
author_facet	Gut, Allan 1944-
author_role	aut
author_sort	Gut, Allan 1944-
author_variant	a g ag
building	Verbundindex
bvnumber	BV023096699
classification_rvk	SK 800
classification_tum	MAT 600f
ctrlnum	(OCoLC)254222247 (DE-599)BVBBV023096699
dewey-full	519.2 510
dewey-hundreds	500 - Natural sciences and mathematics
dewey-ones	519 - Probabilities and applied mathematics 510 - Mathematics
dewey-raw	519.2 510
dewey-search	519.2 510
dewey-sort	3519.2
dewey-tens	510 - Mathematics
discipline	Mathematik
discipline_str_mv	Mathematik
edition	Corr. 2. print.
format	Book
fullrecord	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01734nam a2200469 c 4500</leader><controlfield tag="001">BV023096699</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20111102 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">080124s2007 \|\|\|\| 00\|\|\| eng d</controlfield><datafield tag="015" ind1=" " ind2=" "><subfield code="a">04,N32,0556</subfield><subfield code="2">dnb</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780387228334</subfield><subfield code="9">978-0-387-22833-4</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0387228330</subfield><subfield code="9">0-387-22833-0</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)254222247</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV023096699</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakddb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-384</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-91G</subfield><subfield code="a">DE-824</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">519.2</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">510</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 800</subfield><subfield code="0">(DE-625)143256:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">510</subfield><subfield code="2">sdnb</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 600f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Gut, Allan</subfield><subfield code="d">1944-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)110855329</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Probability</subfield><subfield code="b">a graduate course</subfield><subfield code="c">Allan Gut</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Corr. 2. print.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">New York, NY</subfield><subfield code="b">Springer</subfield><subfield code="c">2007</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XXIII, 603 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Springer texts in statistics</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Wahrscheinlichkeitsrechnung</subfield><subfield code="0">(DE-588)4064324-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Wahrscheinlichkeitstheorie</subfield><subfield code="0">(DE-588)4079013-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="655" ind1=" " ind2="7"><subfield code="0">(DE-588)4123623-3</subfield><subfield code="a">Lehrbuch</subfield><subfield code="2">gnd-content</subfield></datafield><datafield tag="655" ind1=" " ind2="4"><subfield code="a">Lehrbuch - Wahrscheinlichkeitstheorie</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Wahrscheinlichkeitsrechnung</subfield><subfield code="0">(DE-588)4064324-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Wahrscheinlichkeitstheorie</subfield><subfield code="0">(DE-588)4079013-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Augsburg</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016299494&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-016299494</subfield></datafield></record></collection>
genre	(DE-588)4123623-3 Lehrbuch gnd-content Lehrbuch - Wahrscheinlichkeitstheorie
genre_facet	Lehrbuch Lehrbuch - Wahrscheinlichkeitstheorie
id	DE-604.BV023096699
illustrated	Not Illustrated
index_date	2024-07-02T19:43:18Z
indexdate	2024-07-09T21:10:56Z
institution	BVB
isbn	9780387228334 0387228330
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016299494
oclc_num	254222247
open_access_boolean
owner	DE-384 DE-20 DE-11 DE-91G DE-BY-TUM DE-824
owner_facet	DE-384 DE-20 DE-11 DE-91G DE-BY-TUM DE-824
physical	XXIII, 603 S.
publishDate	2007
publishDateSearch	2007
publishDateSort	2007
publisher	Springer
record_format	marc
series2	Springer texts in statistics
spelling	Gut, Allan 1944- Verfasser (DE-588)110855329 aut Probability a graduate course Allan Gut Corr. 2. print. New York, NY Springer 2007 XXIII, 603 S. txt rdacontent n rdamedia nc rdacarrier Springer texts in statistics Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Lehrbuch - Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016299494&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Gut, Allan 1944- Probability a graduate course Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd
subject_GND	(DE-588)4064324-4 (DE-588)4079013-7 (DE-588)4123623-3
title	Probability a graduate course
title_auth	Probability a graduate course
title_exact_search	Probability a graduate course
title_exact_search_txtP	Probability a graduate course
title_full	Probability a graduate course Allan Gut
title_fullStr	Probability a graduate course Allan Gut
title_full_unstemmed	Probability a graduate course Allan Gut
title_short	Probability
title_sort	probability a graduate course
title_sub	a graduate course
topic	Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd
topic_facet	Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie Lehrbuch Lehrbuch - Wahrscheinlichkeitstheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016299494&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT gutallan probabilityagraduatecourse

Holdings

There is no print copy available.

Interlibrary loan Place Request Caution: Not in THWS collection! Indexes

MARC

Record in the Search Index

There is no print copy available.

Similar Items