Lacey received his PhD from the University of Illinois at Urbana-Champaign in 1987, under the direction of Walter Philipp.
[2] His thesis was in the area of probability in Banach spaces, and solved a problem related to the law of the iterated logarithm for empirical characteristic functions.
In the intervening years, his work has touched on the areas of probability, ergodic theory, and harmonic analysis.
While at UNC, Lacey and Walter Philipp gave their proof of the almost sure central limit theorem.
This transform was at the time the subject of a conjecture by Alberto Calderón that Lacey and Christoph Thiele solved in 1996, for which they were awarded the Salem Prize.