An unexpected encounter with Cauchy and Lévy

Natesh S. Pillai, Xiao-Li Meng
2016 Annals of Statistics  
The Cauchy distribution is usually presented as a mathematical curiosity, an exception to the Law of Large Numbers, or even as an "Evil" distribution in some introductory courses. It therefore surprised us when Drton and Xiao [Bernoulli 22 (2016) 38-59] proved the following result for m = 2 and conjectured it for m ≥ 3. Let X = (X 1 , . . . , X m ) and as long as w = (w 1 , . . . , w m ) is independent of (X, Y ), w j ≥ 0, j = 1, . . . , m, and m j =1 w j = 1. In this note, we present an
more » ... e present an elementary proof of this conjecture for any m ≥ 2 by linking Z to a geometric characterization of Cauchy(0, 1) given in Willams [Ann. Math. Stat. 40 (1969) 1083-1085. This general result is essential to the large sample behavior of Wald tests in many Tribute: Like many statisticians, we first learned of the name Peter Hall through his many publications. But one of us (Meng) also had an opportunity to work with him directly, and below is a story from this experience. "When I was a co-editor of Statistica Sinica, I had the great fortune to have Peter serving as an AE. It was a great fortune because one of (any) editor's main tasks is to ensure that each submission is handled promptly. Being a preeminent figure of our profession and with such a prolific research output, Peter had no fewer excuses than any of us to not let his AE ship be his highest priority. It therefore surprised me the first time I asked him to handle a submission. I sent him the request close to midnight. By the time I got on my computer the next morning, a full report from Peter was in, detailing his reasons why the paper did not merit a full review! I never knew what time zone he was on, or whether he had ever slept, but it was a recurring theme that he was the minimal order statistics in terms of response time, yet his report was never rushed in its reasoning or judgement. Eventually I realized that this must be one of the key reasons that he could be the most prolific scholar of our day-always handling whatever came to his desk (or disk) right away, and always with his quick mind fully engaged. It was such an encouragement and inspiration to me for my own editorial work-what excuses did I have not to give it high priority, as I was only publishing a fraction of what Peter published?" Peter was one of those among us whose human capacity cannot be modeled adequately by the ordinary normal model, but rather by Cauchy or even Levy. We therefore thank the editor Runze Li for giving us this opportunity to dedicate this article to Peter in his memory. Peter, you will be sorely missed, not the least by all the authors who wait anxiously for the review reports on their submissions.
doi:10.1214/15-aos1407 fatcat:npnx5bmjmnbf3nah6ycjrtw2ry