# Lecture 11. Doob’s martingale maximal inequalities.

Basic tail and concentration bounds 2 In a variety of settings, it is of interest to obtain bounds on the tails of a random 3 variable, or two-sided inequalities that guarantee that a random variable is close to its 4 mean or median. In this chapter, we explore a number of elementary techniques for 5 obtaining both deviation and concentration inequalities. It is an entrypoint to more 6.

## Kontorovich, Ramanan: Concentration inequalities for.

The way I want to do this is through Azuma's inequality (or any other concentration inequality). The other information I have is that the martingale as defined above, according to Doom's convergence theorem, converge to some random variable a.s. which is something that can be used to define the concentration inequalities but I am not sure about that.We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and web graphs. Article information. Source Internet Math., Volume 3, Number 1 (2006), 79-127. Dates First available in Project Euclid.The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, bounds are obtained on martingale differences associated with the random sequences, which may be of independent interest. As applications of the main.

Sadly, the proof in the original paper (Lemma 6.5) was wrong and as I've found out one can't adapt it after having corrected the wrong inequality (Jensen's inequality used in the wrong direction). The only possibility I see so far is using the Martingale Central limit theorem and then try to control the speed of convergence for my original process using the inequalities described in the.The entire wikipedia with video and photo galleries for each article. Find something interesting to watch in seconds. Martingale techniques Martingales are a central tool in probability theory. In this chapter we illustrate their use, as well as some related concepts, on a number of applications in discrete probability. We begin with a quick review. 3.1 Background To be written. See (Dur10, Sections 4.1, 5.2, 5.7). 3.1.1 Stopping times 3.1.2 .Markovchains: exponentialtailofhittingtimes,andsomecover time. We present the Bernstein-type inequality for widely dependent random variables. By using the Bernstein-type inequality and the truncated method, we further study the strong consistency of estimator of fixed design regression model under widely dependent random variables, which generalizes the corresponding one of independent random variables. As an application of our result, we also derive a concentration inequality for inhomogeneous Markov chains, and establish an extremal property associated with their martingale difference bounds. Proving this requires one to use the anti-concentration inequality for sums of independent random variables. I In the previous problem, it is also true that E(Z n) C 0 logn, but proving it requires anti-concentration inequalities proved very recently and that lie beyond the scope of these lectures. Towards the end, we shall mention this and other anti-concentration inequalities, mostly open. 5. INI Seminar Room 1. Event: (CSMW02) Markov-Chain Monte Carlo Methods.

## Concentration Inequalities for Dependent Random Variables. We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time version of the law of the iterated logarithm, and shed light on the relationship. Matrix Martingales in Randomized Numerical Linear Algebra Speaker Rasmus Kyng Harvard University Sep 27, 2018. The second chapter deals with classical concentration inequalities for sums of independent random variables such as the famous Hoeffding, Bennett, Bernstein and Talagrand inequalities. Further results and improvements are also provided such as the missing factors in those inequalities. The third chapter concerns concentration inequalities for martingales such as Azuma-Hoeffding, Freedman and. In Section 3, we again use the martingale inequality and the symmetry, and provide a tail bound for the independence number. Compared to the isoperimetric inequality and the log-Sobolev inequality, the martingale inequality is rather elementary. Therefore, if one knows the symmetry argument in this paper, in many cases one can use the. When considering uniform martingale concentration over all times without an explicit union bound, the basic tools are Doob’s maximal inequality for nonnegative supermartingales ((), Exercise 5.7.1), Hoeffding’s maximal inequality (), and Freedman’s Bernstein-type inequality ().These can all be easily proved with the techniques of this manuscript (similar to the proof of Theorem 10).

## A large-deviation inequality for vector-valued martingales. A Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration.It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time. The martingale method and its application to the Boolean cube.26 8. Concentration in the product spaces. An application: the law of large numbers for bounded functions.32 9. How to sharpen martingale inequalities?.35 10. Concentration and isoperimetry.39 11. Why do we care: some examples leading to discrete measure concentration questions.44 12. Large deviations and. Get the latest machine learning methods with code. Browse our catalogue of tasks and access state-of-the-art solutions. Tip: you can also follow us on Twitter. It isnoted that the Azuma-Hoeffding inequality for a bounded martingale-difference sequence was extended to centeringsequences with bounded differences (54); this extension provides sharper concentration results for, e.g., sequences that are related to sampling without replacement. The use of the Azuma-Hoeffding inequality was introduced tothe computer science literature in (71) in order to.