New-type Hoeffding’s inequalities and application in tail bounds

It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz [<a href="#1">1</a>]. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refined version by Hertz only considered the first order moment of random variables. In this paper, we present a new type of Hoeffding's inequalities, where the high order moments of random variables are taken into account. It can get some considerable improvements in the tail bounds evaluation compared with the known results. It is expected that the developed new type Hoeffding's inequalities could get more interesting applications in some related fields that use Hoeffding's results.

Probability Bounds for Overlapping Coalition Formation

In this work, we provide novel methods which benefit from obtained probability bounds for assessing the ability of teams of agents to accomplish coalitional tasks. To this end, our first method is based on an improvement of the Paley-Zygmund inequality, while the second and the third ones are devised based on manipulations of the two-sided Chebyshev’s inequality and the Hoeffding’s inequality, respectively. Agents have no knowledge of the amount of resources others possess; and hold private Bayesian beliefs regarding the potential resource investment of every other agent. Our methods allow agents to demand that certain confidence levels are reached, regarding the resource contributions of the various coalitions. In order to tackle real-world scenarios, we allow agents to form overlapping coalitions, so that one can simultaneously be part of a number of coalitions. We thus present a protocol for iterated overlapping coalition formation (OCF), through which agents can complete tasks that grant them utility. Agents lie on a social network and their distance affects their likelihood of cooperation towards the completion of a task. We confirm our methods’ effectiveness by testing them on both a random graph of 300 nodes and a real-world social network of 4039 nodes.

Introduction to Probability and Random Variables

This chapter provides an introduction to probability and random variables. Probability theory is an attempt to formalize the notion of uncertainty in the outcome of an experiment. For instance, suppose an urn contains four balls, colored red, blue, white, and green respectively. Suppose we dip our hand in the urn and pull out one of the balls “at random.” What is the likelihood that the ball we pull out will be red? The chapter first defines a random variable and probability before discussing the function of a random variable and expected value. It then considers total variation distance, joint and marginal probability distributions, independence and conditional probability distributions, Bayes' rule, and maximum likelihood estimates. Finally, it describes random variables assuming infinitely many values, focusing on Markov and Chebycheff inequalities, Hoeffding's inequality, Monte Carlo simulation, and Cramér's theorem.