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Notes + Expository Papers

Expository Papers

Regression Diagnostics for Linear Models: Prepared for Prof. A. Agresti's course on linear and generalized linear models, primarily focusing on Cook's distance and its generalizations via empirical influence functions.

Notes

While both taking and teaching courses, I have prepared notes and expository writings for my own study purposes and to serve as lecture notes. They are posted below in hopes that future students may find them useful. As a result, the notes emphasize intuition over formalism, and are often informal rather than precise.

Group Theory: Currently incomplete lecture notes on group theory, starting from foundations and building up to group operations and representations. Starts with material based on Math 122 course by Prof. B. Gross, with additional material from M. Artin's text (Algebra).

Introduction to Probability Theory: Lecture and section notes based on Stat 110 course at Harvard, taught by Prof. J. Blitzstein and K. Rader, prepared for students as review material

Probability + Measure Theory: Based on combination of 6.436 course at MIT (Prof. D. Gamarnik, with notes by Prof. D. Bertsekas), D. Williams' excellent text (Probability with Martingales, 2013), and text by G. Grimmett and D. Stirzaker (Probability and Random Processes, 2009)

Bayesian Data Analysis: Outline, derivations, and notes to clarify main topics in Bayesian analysis; topics loosely based on Stat 220 course at Harvard (Prof. J. Liu); material based on Gelman et al. (BDA, 2014) and Hoff (A First Course in Bayesian Statistical Methods, 2010)

Linear and Generalized Linear Models: Based on Prof. A. Agresti's course at Harvard (Stat 244) and text (Foundations of Linear and Generalized Linear Models, 2014).

Short Notes & Tutorials

Introduction to R: Prepared for students of Stat 102, 107, 111, and 139 at Harvard.

Markov Chains and Stationarity: Written as a concise and definite introduction to definitions of Markov chain properties and results for stationarity.

The Kelly Formula: A short introduction to the Kelly formula for bet-hedging based on information-theoretic considerations, with introduction to Lagrange multipliers as foundation.

Black-Scholes Equation: Notes on the discrete version of the Black-Scholes pricing formula, inspired by the approach of Terence Tao.

Basic Market Design: Brief notes on market design, motivated by and prepared for lecture given by Scott Kominers for Mathematical Modeling.

Advice on Texts

For most mathematical subjects, the choice of a good text is often one of the biggest hurdles while also, if successful, a major first step in understanding the topic. Personally, as a text-oriented learner, I've had the opportunity/misfortune to peruse a large number of texts in search of one that fit my tastes; for my benefit and potentially a similar reader's, I provide some references for major subjects.

Probability: After a near-exhaustive search through the literature, I finally stumbled upon Williams' Probability with Martingales and Pollard's User's Guide to Measure-Theoretic Probability as the gems in the field. Unfortunately, they are not particularly well-suited for first-time learners, or those without a second course in real analysis (i.e. measure theory, elementary functional analysis).

Statistics: Inevitably, one has to take a pass through Casella/Berger's Statistical Inference, and despite many gripes, it is generally one of the best in the field. I would supplement with Wasserman's All of Statistics for a wider, bird's-eye view (which is often lost in the former text) or Davison's Statistical Models for an even grittier one.

Real Analysis: No matter the time, Rudin's trio cannot be beat. I'm no big fan of Baby Rudin (Principles of Mathematical Analysis) but a huge one of Papa Rudin (Real and Complex Analysis). The third ("Grandpa"?) Rudin text on Functional Analysis is much more taste-dependent. For areas such as operator theory, I've found Douglas' Banach Algebra Techniques in Operator Theory to be a very readable yet rigorous account.

Stochastic Processes: Depending on your interests, I would start with Rogers/Williams' Diffusions, Markov Processes, and Martingales, potentially moving to Stroock/Varadhan's Multidimensional Diffusion Processes. For stochastic integration, Chung/Williams' Introduction to Stochastic Integration appears excellent. For Markov processes, Ethier/Kurtz's book of the same name deserves merit.

Convergence of Measures: While not a widely-read area, it is very useful in studying theoretical MC methods or particle methods; standard texts are by Parthasarathy (Probability Measures on Metric Spaces) and Billinsley (Convergence of Probability Meausres).