Convergence of Probability Measures
by Patrick Billingsley · first published 1968
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A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. Widely known for his straightforward approach and reader-friendly style, Dr. Billingsley presents a clear, precise, up-to-date account of probability limit theory in metric spaces. He incorporates many examples and applications that illustrate the power and utility of this theory in a range of disciplines-from analysis and number theory to statistics
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…0-12-079240-0, Academic Press, New York.\ \ Robert J.\ Serfling, {\it Approximation Theorems of Mathematical Statistics,\/} ISBN 0-471-02403-1, John Wiley and Sons, New York.\ \ P.\ Billingsley, {\it Convergence of Probability Measures, 2\raise0.5ex\hbox{ed},\/} ISBN: 0-471-19745-9, John Wiley, New York.\ \ R.\ S.\ Lipster and A.\ N.\ Shiryayev, {\it Statistics of Random Processes I, II,} ISBN 0-387-90226-0, Springer-Verlag, New York.\…
on “Ask HN: I want to learn statistics and data mining” ↗…partner on their own can find the typo. Should such a person wish to try, then reply here, and I will send a PDF of the paper. Hint: It would be good to be able to find, say, Patrick Billingsley, 'Convergence of Probability Measures' fun and easy reading, and for that should have greatly enjoyed, say, at least the first (real) half of Walter Rudin, 'Real and Complex Analysis' and Walter Rudin, 'Principles of Mathematical Analysis'.…
on “First Round Capital Open Application for Startups” ↗…lectures at Stanford. We've known a lot about pure and applied statistics and how to use them for a very long time. The good work in statistics is high quality pure/applied math, e.g., Billingsly, 'Convergence of Probability Measures', e.g. with Ulam's result Le Cam called 'tightness', Brillinger on time series, Serfling on limit theorems, Rao on linear methods, Breiman on CART, and much more. I'm not seeing comparable quality in…
on “Basics of Machine Learning” ↗…significance of the test, and used a classic result, LeCam called "tightness" of S. Ulam to show that the power of the test was better than that of a trivial test. Ulam's result is in P. Billingsley, Convergence of Probability Measures. Billingsley was long at U. Chicago.
on “Outlier Detection Techniques (2010) [pdf]” ↗…try, his first contribution yielded about 15 thousand tons. Before that he had a nice result in probability the French probabilist Le Cam called 'tightness' with a nice presentation in Billingsley's 'Convergence of Probability Measures'. (2) The Research I can believe that the VCs don't receive many promising proposals based on research. One reason is that not many talented, well trained researchers cross over to become entrepreneurs.…
on “Ask HN: What are the most innovative VC/Angel funds out there?” ↗…that the probability mass farther than r from the origin is less than h. This result holds in spaces of considerable generality, but I'd have to look up the details now. Details are in P. Billingsley, 'Convergence of Probability Measures', 1999. I used the result in a paper once. It's nice result and gets used occasionally in advanced work in probability.
on “Ulam spiral” ↗…Billingsley, and finally Durrett. I used Feller I and/or II for reference. Otherwise his writing seemed to lack an overall unification. The only Billingsley text I used, and then only for a small topic, was Convergence of Probability Measures.
on “Foundations of probability theory” ↗…space in Walter Rudin, Real and Complex Analysis. There are some nice applications of the Hahn-Banach theorem in David G. Luenberger, Optimization by Vector Space Methods. In Patrick Billingsley, Convergence of Probability Measures. is a nice presentation of Ulam's result in measure theory Le Cam called tightness: Roughly, IIRC, for any probability measure P and any a > 0 no matter how small, there exists a sphere S of finite…
on “The Eccentric Lives of Steinhaus, Banach and Ulam (2014)” ↗As a very general but crude and blunt approach to show that the hypotheses tests were not trivial, used the result of S. Ulam that Le Cam called "tightness" as in P. Billingsley, Convergence of Probability Measures. When doing both multi-dimensional and distrbution-free, are nearly way out in the ozone so get pushed into some abstract techniques! Meow!
on “Foundations Machine Learning” ↗Ulam has a result in measure theory the French mathematician LeCam called tightness. The result is in Patrick Billingsley, Convergence of Probability Measures, John Wiley and Sons, New York, 1968. The result was used in a course I took in measure theory based probability theory. Later I used the result in one paper to show that a goofy statistical hypothesis…
on “Adventures of a Mathematician brings an unsung scientist back into the light” ↗…some. I used some group theory, summed used the classic S. Ulam, guy on the left in http://www-history.mcs.st-and.ac.uk/BigPictures/Ulam_Feynman... result LeCam called tightness (see P. Billingsley, Convergence of Probability Measures), and was able to permit selecting false alarm rate in advance and then getting that rate exactly. Asking for all of the Neyman-Pearson result would take more data than we had any chance of having, but…
on “The 1980 Math Olympiad Program: Where are they now?” ↗…probability, and some work in stochastic processes. At one point I use a classic result of S. Ulam that the French probabilist Le Cam called tightness -- there is a nice presentation in Patrick Billingsley, Convergence of Probability Measures, John Wiley and Sons, New York, 1968. The work was for anomaly detection in complex systems so could be regarded as for both engineering and machine learning. But I can't give all the background here.
on “Introduction to Probability at an advanced level (2018) [pdf]” ↗…anomaly detection, both multi-variate and distribution-free. And there I used the S. Ulam result tightness that even in the medical statistics community is likely not well known (it's in P. Billingsley, Convergence of Probability Measures). So, I have some technical qualifications to talk about test quality. But here I'm not trying to address, comment on, inform, educate, or critique the medical profession. Instead: My real point is,…
on “Ask HN: Are we overreacting to coronavirus?” ↗…alarm rate. And can use a classic result in measure theory by S. Ulam, that the French probabilist LeCam nicely called 'tightness', to show that the hypothesis test is not 'trivial'. See Billingsly 'Convergence of Probability Measures'. Of course, the 'most powerful' test for each given false alarm rate is from the Neyman-Pearson lemma (the elementary proofs are not much fun, but there is a nice proof starting with the Hahn decomposition,…
on “What I wish systems researchers would work on” ↗