Welcome to the probability inequality mines!
When something in your process (measurement, estimation) means that you can be pretty sure that a whole bunch of your stuff is damn likely to be somewhere in particular.
This is basic workhorse stuff in univariate probability, and turns out to be yet more essential in multivariate matrix probability, as seen in matrix factorisation, compressive sensing, PAC-bounds and suchlike.
- this super simple intro to chaining and controlling maxima by Thomas Lumley
- Dasgupta, Asymptotic Theory of Statistics and Probability (Dasg08) is very easy, and despite its name introduces some nice basic non-asymptotic inequalities
- Raginsky and Sason, Concentration of Measure Inequalities in Information Theory, Communications and Coding (RaSa12)
- Tropp, An Introduction to Matrix Concentration Inequalities (Trop15) high-dimensional data! free!
- Boucheron, Bousquet & Lugosi, Concentration inequalities (BoBl04a) (Clear and brisk but missing some newer stuff)
- Massart, Concentration inequalities and model section (Mass07). Clear, and focussed, but very quick and further, depressingly, by being applied it also demonstrates the limitations of these techniques. Mass00 is an earlier draft.
- Boucheron, Lugosi & Massart, Concentration inequalities: a nonasymptotic theory of independence (BoLM13). Haven't read it yet.
Lugosi's Concentration-of-measure Lecture notes:
The inequalities discussed in these notes bound tail probabilities of general functions of independent random variables.
The taxonomy is interesting:
Several methods have been known to prove such inequalities, including martingale methods (see Milman and Schechtman  and the surveys of McDiarmid [2, 3]), information-theoretic methods (see Alhswede, Ga ́cs, and Ko ̈rner , Marton [5, 6, 7], Dembo , Massart  and Rio ), Talagrand’s induction method [11, 12, 13] (see also Luczak and McDiarmid , McDiarmid  and Panchenko [16, 17, 18]), the decoupling method surveyed by de la Pen ̃a and Gin ́e , and the so-called “entropy method”, based on logarithmic Sobolev inequalities, developed by Ledoux [20, 21], see also Bobkov and Ledoux , Massart , Rio , Klein , Boucheron, Lugosi, and Mas- sart [25, 26], Bousquet [27, 28], and Boucheron, Bousquet, Lugosi, and Massart .
(actioned in his Combinatorial statistics notes)
Foundational but impenetrable things I won't read right now: Talagrand's opus that is commonly credited with kicking off the modern fad especially with the chaining method. (Tala95)
Finite sample bounds
These are everywhere in statistics. Special attention will be given here to finite-sample inequalities. Asymptotic normality is so last season. These days we care about finite sample performance, and asymptotic results don't help us there. Apparently I can construct useful bounds using concentration inequalities? One suggested keyword to disambiguate: Ahlswede-Winterfeld bounds?
For the Gaussian distribution. Filed there.
Let us copy from wikipedia:
Heuristically: if we pick complex numbers , and add them together, each multiplied by jointly independent random signs , then the expected value of the sum's magnitude is close to .
Let i.i.d. random variables with for , i.e., a sequence with Rademacher distribution. Let and let . Then
for some constants . It is a simple matter to see that when , and when )0 < p \le 2).
Empirical process theory
Large deviation inequalities, empirical process inequalities, Talagrand chaining method. Berry-Esseen bound.
Matrix Chernoff bounds
Nikhil Srivastava's Discrepancy, Graphs, and the Kadison-Singer Problem has an interesting example of bounds via discrepancy theory (and only indirectly probability). Gros11 is also readable, and gives quantum-mechanical results (i.e. the matrices are complex-valued).
In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic.
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