Deviation Inequalities in Probability Theory

The laws of large numbers say that averages taken over large samples converge on expectation values. But these are asymptotic statements which say nothing about what happens for samples of any particular size. A deviation inequality, by contrast, is a result which says that, for realizations of such-and-such a stochastic process, the sample value of this functional deviates by so much from its typical value with no more than a certain probability: \( \Pr{\left(\left|f(X_1, X_2, \ldots X_n) - \mathbb{E}[f]\right| > h \right)} < r(n,h,f) \), where the rate function \( r \) has to be given explicitly, and may depend on the true joint distribution of the \( X_i \) (though it's more useful if it doesn't depend on that very much). (And of course one could compare to the median rather than the mean, or just look at fluctuations above the typical value rather than to other side, or get within a certain factor of the typical value rather than a certain distance, etc.) The rate should be decreasing in \( h \) and in \( n \).

An elementary example is "Markov's inequality": If \( X \) is a non-negative random variable with a finite mean, then \[ \Pr{\left(X > h \right)} < \frac{\mathbb{E}[X]}{h} ~. \] One can derive many other deviation inequalities from Markov's inequality by taking \( X = g(Y) \), where \( Y \) is another random variable and \( g \) is some suitable non-negative-valued function.

For instance if \( Y \) has a finite variance \( v \), then \[ \Pr{\left(|Y-\mathbb{E}[Y]| > h \right)} < \frac{v}{h^2} ~. \] This is known as "Chebyshev's inequality". (Exercise: derive Chebyshev's inequality from Markov's inequality. Since Markov was in fact Chebyshev's student, it would seem that the logical order here reverses the historical one, though guessing at priority from eponyms is always hazardous.) Suppose that \( X_1, X_2, \ldots \) are random variables with a common mean \( m \) and variance \( v \), and \( Y \) is the average of the first \( n \) of these. Then Chebyshev's inequality tells us that \[ \Pr{\left(|Y - m| > h\right)} < \frac{\mathrm{Var}[Y]}{h^2} ~. \] If the \( X_i \) are uncorrelated (e.g., independent), then \( \mathrm{Var}[Y] = v/n \), so the probability that the sample average differs from the expectation by \( h \) or more goes to zero, no matter how small we make \( h \). This is precisely the weak law of large numbers. If the \( X_i \) are correlated, but nonetheless \( \mathrm{Var}[Y] \) goes to zero as \( n \) grows (generally because correlations decay), then we get an ergodic theorem. The rate of convergence here however is not very good, just \( O(n^{-1}) \).

Since \( e^u \) is a monotonically increasing function of \( u \), for any positive \( t \), \( X > h \) if and only if \( e^{tX} > e^{th} \), so we get an exponential inequality, \[ \Pr{\left(X > h\right)} < e^{-th} \mathbb{E}\left[e^{tX}\right] ~. \] Notice that the first term in the bound does not depend on the distribution of \( X \), unlike the second term, which doesn't depend on the scale of the deviation \( h \). We are in fact free to pick whichever \( t \) gives us the tightest bound. The quantity \( \mathbb{E}\left[e^{tX}\right] \) is called the "moment generating function" of \( X \), let's abbreviate it \( M_X(t) \), and can fail to exist if some moments are infinite. (Write the power series for \( e^u \) and take expectations term by term to see all this.) It has however the very nice property that when \( X_1 \) and \( X_2 \) are independent, \( M_{X_1+X_2}(t) = M_{X_1}(t) + M_{X_2}(t) \). From this it follows that if \( Y \) is the sum of \( n \) independent and identically distributed copies of \( X \), \( M_Y(t) = {(M_X(t))}^{n} \). Thus \[ \Pr{\left(Y > h\right)} < e^{-th} (M_X(t))^{n} ~. \] If \( Z = Y/n \), the sample mean, this in turn gives \[ \Pr{\left(Z > h\right)} = \Pr{\left(Y > hn\right)} < e^{-thn} (M_X(t))^{n} ~. \] So we can get exponential rates of convergence for the law of large numbers from this. [Students who took the CMU statistics department's probability qualifying exam in 2010 now know who wrote problem 9.] Again, the restriction to IID random variables is not really essential, allowing dependence just means that the moment generating functions don't factor exactly, but if they almost factor than we can get results of the same form. (Often, we end up with \( n \) being replaced by \( n/n_0 \), where \( n_0 \) is something like how long it takes dependence to decay to trivial levels.)

I don't feel like going into the reasoning behind the other common deviation bounds — Bernstein, Chernoff, Hoeffding, Azuma, McDiarmid, etc. — because I feel like I've given enough of the flavor already. I am using this notebook as, actually, a notebook, more specifically a place to collect references on deviation inequalities, especially ones that apply to collections of dependent random variables. Results here typically appeal to various notions of mixing or decay of correlations, as found in ergodic theory.

G. G. Bosco, F. P. Machado and Thomas Logan Ritchie, "Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach", Journal of Statistical Physics 139 (2010): 367--374
Olivier Bousquet, Stéphane Boucheron and Gábor Lugosi, "Introduction to Statistical Learning Theory" [Gives a very nice review of many deviation inequalities, with references.]
Nicolo Cesa-Bianchi and Gabor Lugosi, Prediction, Learning, and Games [Provides exemplary proofs in the appendix. Mini-review]
J.-R. Chazottes, P. Collet, C. Kuelske and F. Redig, "Deviation inequalities via coupling for stochastic processes and random fields", math.PR/0503483 [Very cool]
Iosif Pinelis, "Between Chebyshev and Cantelli", arxiv:1011.6065 [Cute, with an appealing proof]
Yongqiang Tang, "A Hoeffding-Type Inequality for Ergodic Time Series", Journal of Theoretical Probability 20 (2007): 167--176 [PDF preprint]

Olivier Catoni
- "High confidence estimates of the mean of heavy-tailed real random variables", arxiv:0909.5366
- "Challenging the empirical mean and empirical variance: a deviation study", arxiv:1009.2048
Patrick Cattiaux and Arnaud Guillin, "Deviation bounds for additive functionals of Markov process", math.PR/0603021 [Non-asymptotic bounds for the probability that time averages deviate from expectations with respect to the invariant measure, when the process is stationary and ergodic and the invariant measure is reasonably regular.]
Jérôme Dedecker, Florence Merlevède, Magda Peligrad, Sergey Utev, "Moderate deviations for stationary sequences of bounded random variables", arxiv:0711.3924
Marian Grendar Jr. and Marian Grendar, "Chernoff's bound forms," math.PR/0306326
A. Guionnet and B. Zegarlinski, Lectures on Logarithmic Sobolev Inequalities [120 pp. PDF]
Vladislav Kargin, "A large deviation inequality for vector functions on finite reversible Markov Chains", Annals of Applied Probability 17 (2007): 1202--1221, arxiv:0508538
Carlos A. Leon and Francois Perron, "Optimal Hoeffding bounds for discrete reversible Markov chains", math.PR/0405296
Dasha Loukianova, Oleg Loukianov, Eva Loecherbach, "Polynomial bounds in the Ergodic Theorem for positive recurrent one-dimensional diffusions and integrability of hitting times", arxiv:0903.2405 [non-asymptotic deviation bounds from bounds on moments of recurrence times]
P. Major
- "On a multivariate version of Bernstein's inequality", math.PR/0411287
- "A multivariate generalization of Hoeffding's ineqality", math.PR/0411288
Florence Merlevède, Magda Peligrad, Emmanuel Rio , "A Bernstein type inequality and moderate deviations for weakly dependent sequences", Probability Theory and Related Fields 151 (2011): 435--474, arxiv:0902.0582
Ted Theodosopoulos, "A Reversion of the Chernoff Bound", math.PR/0501360

(Thanks to Michael Kalgalenko for typo-spotting)

Notebooks

Deviation Inequalities in Probability Theory