When I teach tail bounds, I use the usual progression:

  • If your r.v is positive, you can apply Markov's inequality
  • If you have independence and also bounded variance, you can apply Chebyshev's inequality
  • If each independent r.v also has all moments bounded, then you can use a Chernoff bound.

After this things get a little less clean. For example

  • If your variables have zero mean, then a Bernstein inequality is more convenient
  • If all you know is that the combining function is Lipschitz, then there's a generalized McDiarmid-style inequality
  • if you have weak dependence then there are Siegel-style bounds, (and if you have negative dependence, then Jansson's inequality might be your friend)

Is there a reference anywhere to a convenient flowchart or decision tree describing how to choose the "right" tail bound, (or even when you have to dive into a sea of Talagrand) ?

I'm asking partly so that I have a reference, partly so that I can point it to my students, and partly because if I'm sufficiently annoyed and there isn't one, I might try to make one myself.

  • $\begingroup$ I think the simple answer is no and yes please to anyone who makes one. $\endgroup$
    – Simd
    Commented Jun 23, 2014 at 12:52

1 Answer 1


Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey available at http://projecteuclid.org/euclid.im/1175266369 or at Fan Chung Graham's web page.

  • $\begingroup$ Yes ! this is excellent ! I've read this survey before, but completely forgot about it. $\endgroup$ Commented Jun 20, 2014 at 11:04
  • 6
    $\begingroup$ It's a very nice survey, but I don't see anything like what is requested in the original post: "a convenient flowchart or decision tree describing how to choose the 'right' tail bound" for the random variables you have. $\endgroup$
    – usul
    Commented Jun 20, 2014 at 13:25
  • $\begingroup$ It's not exactly right, but there are flow charts showing how the different theorems imply each other, which is a start. $\endgroup$ Commented Jun 23, 2014 at 16:08

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