If X is a martingale, using both inequalities above and applying the
union bound
allows one to obtain a two-sided bound:
Proof
The proof shares similar idea of the proof for the general form of Azuma's inequality listed below. Actually, this can be viewed as a direct corollary of the general form of Azuma's inequality.
A general form of Azuma's inequality
Limitation of the vanilla Azuma's inequality
Note that the vanilla Azuma's inequality requires symmetric bounds on martingale increments, i.e. . So, if known bound is asymmetric, e.g. , to use Azuma's inequality, one need to choose which might be a waste of information on the boundedness of . However, this issue can be resolved and one can obtain a tighter probability bound with the following general form of Azuma's inequality.
Statement
Let be a martingale (or supermartingale) with respect to filtration. Assume there are predictable processes and with respect to , i.e. for all , are -measurable, and constants such that
almost surely. Then for all ,
Since a submartingale is a supermartingale with signs reversed, we have if instead is a martingale (or submartingale),
If is a martingale, since it is both a supermartingale and submartingale, by applying union bound to the two inequalities above, we could obtain the two-sided bound:
Proof
We will prove the supermartingale case only as the rest are self-evident. By Doob decomposition, we could decompose supermartingale as where is a martingale and is a nonincreasing predictable sequence (Note that if itself is a martingale, then ). From , we have
Finally, since and as is nonincreasing, so event implies , and therefore
Remark
Note that by setting , we could obtain the vanilla Azuma's inequality.
Note that for either submartingale or supermartingale, only one side of Azuma's inequality holds. We can't say much about how fast a submartingale with bounded increments rises (or a supermartingale falls).
Simple example of Azuma's inequality for coin flips
Let Fi be a sequence of independent and identically distributed random coin flips (i.e., let Fi be equally likely to be −1 or 1 independent of the other values of Fi). Defining yields a martingale with |Xk − Xk−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get
For example, if we set t proportional to n, then this tells us that although the maximum possible value of Xn scales linearly with n, the probability that the sum scales linearly with ndecreases exponentially fast with n.
If we set we get:
which means that the probability of deviating more than approaches 0 as n goes to infinity.
Hoeffding proved this result for independent variables rather than martingale differences, and also observed that slight modifications of his argument establish the result for martingale differences (see page 9 of his 1963 paper).
^It is not a direct application of Hoeffding's lemma though. The statement of Hoeffding's lemma handles the total expectation, but it also holds for the case when the expectation is conditional expectation and the bounds are measurable with respect to the sigma-field the conditional expectation is conditioned on. The proof is the same as for the classical Hoeffding's lemma.
References
Alon, N.; Spencer, J. (1992). The Probabilistic Method. New York: Wiley.
(1937). О некоторых модификациях неравенства Чебышёва [On certain modifications of Chebyshev's inequality]. Doklady Akademii Nauk SSSR (in Russian). 17 (6): 275–277. (vol. 4, item 22 in the collected works)
McDiarmid, C. (1989). "On the method of bounded differences". Surveys in Combinatorics. London Math. Soc. Lectures Notes 141. Cambridge: Cambridge Univ. Press. pp. 148–188.
Godbole, A. P.; Hitczenko, P. (1998). "Beyond the method of bounded differences". Microsurveys in Discrete Probability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. Vol. 41. pp. 43–58.