Generalizations of some concentration inequalities

Random Structures and Algorithms, 2000

We present a new general concentration-of-measure inequality and illustrate its power by applications in random combinatorics. The results nd direct applications in some problems of learning theory.

Probability Theory and Related Fields, 1990

Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function¢. In the case ¢(x)=(x-c)2, where c is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Levy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.

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 result, concentration inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale difference bounds is established. This work complements and generalizes certain concentration inequalities obtained by Marton and Samson, while also providing different proofs of some known results.

ArXiv, 2019

In this report, we aim to exemplify concentration inequalities and provide easy to understand proofs for it. Our focus is on the inequalities which are helpful in the design and analysis of machine learning algorithms.

Communications in Mathematical Physics, 2012

Using martingale methods, we obtain some Fuk-Nagaev type inequalities for suprema of unbounded empirical processes associated with independent and identically distributed random variables. We then derive weak and strong moment inequalities. Next, we apply our results to suprema of empirical processes which satisfy a power-type tail condition. Abstract Using martingale methods, we obtain some Fuk-Nagaev type inequalities for suprema of unbounded empirical processes associated with independent and identically distributed random variables. We then derive weak and strong moment inequalities. Next, we apply our results to suprema of empirical processes which satisfy a power-type tail condition.

Electronic Journal of Probability, 2009

We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order 1 + ε of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1 + ε is finite uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and nonuniform behavior of moments of the coupling time can occur.

We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffding's inequalities for bounded scalar random variables. As applications, we apply the general results to multinomial and Dirichlet distributions to obtain multivariate concentration inequalities.

Journal of Inequalities and Applications, 2001

The aim of this paper is to show that Jensen's Inequality and an extension of Chebyshev's Inequality complement one another, so that they both can be formulated in a pairing form, including a second inequality, that provides an estimate for the classical one.

arXiv: Probability, 2005

For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussian concentration inequalities, transportation inequalities and also logarithmic Sobolev inequalities in the case of the Euclidean space. In several cases, the obtained constants are of optimal order of growth with respect to the number of variables, or are independent of this number. These results extend results known for mutually independent variables to weakly dependent variables under Dobrushin-Shlosman type conditions.