Jim Pitman paid tribute to his long-term colleague and distinguished researcher in probability theory Michael J. Klass, UC Berkeley, on Michael’s retirement in July 2020. Michael will continue to be involved in the department as Professor Emeritus and Professor in the Graduate School. Below, Jim describes Michael’s career:

Despite having broken his neck and using a wheelchair since 1965, Michael Klass received his PhD in 1972 in Mathematics at UCLA, where his thesis work in enumerative combinatorics was advised by Bruce Rothschild. Subsequently he wrote a paper fully generalizing Burnside’s foundational combinatorial lemma. After a post-doctoral position at Caltech and the Jet Propulsion Laboratory, he was appointed in Berkeley first as a Miller Fellow in 1974, then as Assistant Professor in 1975,  moving through the ranks to Full Professor in 1984.  He was elected a Fellow of the IMS in 1979.

In addition to his thesis work at UCLA in the 1970s, Michael collaborated with Tom Ferguson, leading to the Ferguson–Klass representation of atom sizes in Dirichlet and other completely random measures. These now play a central role in modern Bayesian nonparametric theory, and in associated machine learning algorithms such as latent Dirichlet allocation. In other early work, Michael studied optimal stopping problems for normalized sums of independent random variables, and associated maximal inequalities and limit theorems.

In the mid to late 1970s, Michael developed extensions of the classical law of the iterated logarithm for random variables not subject to traditional moment conditions. That involved developing novel functionals of the distribution of a long-tailed random variable (used to construct expectations of functions of sums of independent random variables), which dictate the correct normalization constants for such a result, and a sustained analysis of how these functionals of the underlying distribution of terms affect the long term fluctuations of their partial sums.

By the early 1980s Michael was widely acknowledged as the world’s leading expert in the theory of fluctuations of random sums under minimal moment conditions. In other work around this time, Michael collaborated on the development of an estimate of volatility for security prices, based on an analysis of Brownian motion, the now widely cited Garman–Klass volatility estimator. In later work on mathematical finance, he collaborated with Michael Taksar and David Assaf, on an influential article about maximizing the rate at which money can be compounded when moving it back and forth between a log normally distributed asset and the bank, in the presence of brokerage fees.

In a series of papers with Marjorie Hahn in the 1980s and 1990s, Michael developed results for matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal, symmetric stable laws and general affine normalization of sums. Their earlier joint work established a condition for continuity of Gaussian processes, and their later work obtained results for the Radon transform for infinite measures, and a local probability approximation for sums of uniformly bounded independent random variables. In addition, they obtained sharp results on the exponential order of tail probabilities of sums of independent and identically distributed random variables. He continued to work in the 1990s on a variety of problems related to maximal inequalities and the rate of growth of partial sums of independent random variables, including best possible forms of Wald’s identity, later demonstrating such results did not extend to randomly stopped quadratic forms of independent random variables.

In the late 1990s, Michael’s interest shifted to the study of self-normalized processes. Such processes are the basis of many statistical methods, dating back to the famous Student’s t-test due to William Gosset in the early 1900s.

Normalized processes also arise in the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and parameter estimation, and Studentized pivots and bootstrap-t methods for confidence intervals.

From the late 1990s until around 2010, Michael worked with Victor de la Peña at Columbia and Tze Leung Lai at Stanford on limit theorems for self-normalized martingales, based on exponential inequalities derived from variants of Wald’s likelihood ratio martingale. The underlying method, of working with integral mixtures of Wald martingales to obtain maximal inequalities, traces back to the 1970 work of Robbins and Siegmund on boundary crossings of Brownian motion. Michael coined the term “pseudo-maximization” for the method, and showed how it could be exploited to obtain new results controlling the behavior of self-normalized martingales. As Michael shows, this technique is very effective in establishing various inequalities for self-normalized processes, in particular exponential bounds and moment bounds, and in the proof of the law of the iterated logarithm, which he showed could be established for arbitrary sums of random variables without any conditions save that the sum diverges. See the 2007 review article of de la Peña, Klass and Lai in Probability Surveys for a masterly exposition of the method and its applications.

In the late 1990s, Michael collaborated with Krzysztof Nowicki on quadratic forms (and their generalizations) of independent random variables, and expectations of functions thereof. Thereafter they obtained a landmark improvement of Hoffmann-Jørgensen’s inequality, somewhat refining and generalizing it later in best possible fashion. Subsequently they made an important contribution to investment theory with limited drawdown. From 2007 until 2016 their work on tail probabilities of arbitrary sums of independent random variables became definitive, ultimately applying without conditions on the random variables or the levels to be exceeded. In addition, they produced a paper with results on optimal growth and distribution of wealth subject to a drawdown constraint, as well as a sequential search algorithm for determining the exact size of a given population. Important coauthored papers include one with Cun Hui Zhang and one with Kwok Pui Choi.

More recently, Michael has begun work on hypothesis testing in statistics.

Over the course of his career at Berkeley, Michael served as the advisor of five PhD students, several of whom went on to teaching careers, and as the teacher of about 4,000 students at Berkeley and the mentor of many of them. One of his students, Victor de la Peña, now at Columbia University, contributed the following impression, which sums up the feelings of Michael’s many students and colleagues over the years, and serves to conclude this brief review of Michael’s career:

“Mike is an exemplary teacher, a profound thinker and a compassionate human being, who has positively impacted the lives of many students and fellow researchers. I particularly consider myself fortunate to have had him as an adviser, collaborator and friend. He has been my mentor starting from the time I was a student at Berkeley, guiding me in every step of my career. He is not only an accomplished mathematician, but also a scholar of the Old Testament (still making unusual contributions) and an avid chess player. I have benefitted greatly from our interactions and discussions about probability, philosophy, and religion. He is very compassionate and eager to share his insights with others. His approach to teaching involves conceptualizing the problem, no matter how difficult it is, breaking complex ideas into digestible parts, and presenting them in the most straightforward manner. He is strong in the face of adversity and resilient beyond any measurable degree. Most importantly, his seriousness of purpose is only matched by his wonderful sense of humor. As Mike starts a new chapter in his illustrious career, we wish him good health and longevity, so that he continues to enrich our lives by his wisdom, camaraderie and scholarship.”