## Probability for StatisticiansProbability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more than half the text, and the choices the author makes at the University of Washington in Seattle are spelled out. The rest of the text provides background, offers different routes that could be pursued in the classroom, ad offers additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic funcion presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. The martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. The author is a professor of Statistics and adjunct professor of Mathematics at the University of Washington in Seattle. He served as chair of the Department of Statistics 1986-- 1989. He received his PhD in Statistics from Stanford University. He is a fellow of the Institute of Mathematical Statistics, and is a former associate editor of the Annals of Statistics. |

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### Contents

12 | |

3 | 37 |

4 | 61 |

The Fundamental Theorem of Calculus | 74 |

General Topology and Hilbert Space | 95 |

Distribution and Quantile Functions 1 Character of Distribution Functions | 107 |

Properties of Distribution Functions | 110 |

The Quantile Transformation | 111 |

Classical Convergence in Distribution | 288 |

Limit Determining Classes of Functions | 292 |

Brownian Motion and Empirical Processes 1 Special Spaces | 295 |

Existence of Processes on CC and DD | 298 |

Brownian Motion and Brownian Bridge | 302 |

Stopping Times | 305 |

Strong Markov Property | 308 |

Embedding a RV in Brownian Motion | 311 |

Integration by Parts Applied to Moments | 115 |

Important Statistical Quantities | 119 |

Infinite Variance | 123 |

Slowly Varying Partial Variance | 127 |

Specific Tail Relationships | 134 |

Regularly Varying Functions | 137 |

Some Winsorized Variance Comparisons | 140 |

Inequalities for Winsorized Quantile Functions | 147 |

Independence and Conditional Distributions 1 Independence | 151 |

The Tail σField | 156 |

Uncorrelated Random Variables | 157 |

Basic Properties of Conditional Expectation | 158 |

Regular Conditional Probability | 168 |

Conditional Expectations as Projections | 174 |

Special Distributions 1 Elementary Probability | 179 |

Distribution Theory for Statistics | 187 |

Linear Algebra Applications | 191 |

The Multivariate Normal Distribution | 199 |

WLLN SLLN LIL and Series 0 Introduction | 203 |

BorelCantelli and Kronecker Lemmas | 204 |

Truncation WLLN and Review of Inequalities | 206 |

Maximal Inequalities and Symmetrization | 210 |

The Classical Laws of Large Numbers LLNs | 215 |

Applications of the Laws of Large Numbers | 223 |

General Moment Estimation | 226 |

Law of the Iterated Logarithm | 235 |

Strong Markov Property for Sums of IID RVs | 239 |

Convergence of Series of Independent RVs | 241 |

Martingales | 246 |

Maximal Inequalities Some with Boundaries | 247 |

A Uniform SLLN | 252 |

Convergence in Distribution 1 Steins Method for CLTs | 255 |

Winsorization and ˇTruncation | 264 |

Identically Distributed RVs | 269 |

Bootstrapping | 274 |

Bootstrapping with Slowly Increasing Trimming | 276 |

Examples of Limiting Distributions | 279 |

Barrier Crossing Probabilities | 314 |

Embedding the Partial Sum Process | 318 |

Other Properties of Brownian Motion | 323 |

Various Empirical Processes | 333 |

Applications | 338 |

Characteristic Functions 1 Basic Results with Derivation of Common Chfs | 341 |

Uniqueness and Inversion | 346 |

The Continuity Theorem | 350 |

Elementary Complex and Fourier Analysis | 352 |

Esseens Lemma | 358 |

Distributions on Grids | 361 |

Conditions for φ to Be a Characteristic Function | 363 |

CLTs via Characteristic Functions Edgeworth Expansions 390 | 365 |

Basic Limit Theorems | 366 |

Variations on the Classical CLT | 371 |

Local Limit Theorems | 380 |

Gamma Approximation | 383 |

Infinitely Divisible and Stable Distributions | 399 |

Stable Distributions | 407 |

Asymptotics via Empirical Proceses | 415 |

Asymptotics via Steins Approach | 449 |

Hoeffdings Combinatorial CLT | 458 |

Martingales | 467 |

The Submartingale Convergence Theorem | 473 |

Applications of the Smg Convergence Theorem | 481 |

Decomposition of a Submartingale Sequence | 487 |

Applications of Optional Sampling | 499 |

DoobMeyer Submartingale Decomposition | 511 |

The Basic Censored Data Martingale | 522 |

CLTs for Dependent RVs | 529 |

Metrics for Convergence in Distribution | 540 |

The Gamma and Digamma Functions | 546 |

Examples of Statistical Models | 555 |

Asymptotics of Maximum Likelihood Estimation | 563 |

575 | |

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### Common terms and phrases

absolutely continuous asymptotic bootstrap Borel Borel sets bounded Brownian bridge Brownian motion called Chapter claim compact condition Consider continuous functions convergence in distribution corollary countable define Definition denote density df F equal equivalent example Exercise 1.1 exists finite fixed gives holds implies independent rvs inequality interval Lebesgue measure lemma Let X1 linear martingale mean measurable function metric space Moreover normal notation Note o-field Poisson probability space Proof proposition quantile random result s-mg sample satisfies sequence Show signed measure SLLN ſº statistics submg subsets Suppose symmetric Theorem 3.1 trivial uniform uniformly integrable variance vector Verify Winsorized WLLN