Given a bag with r red balls and g green balls, suppose that...

Given a bag with $r$ red balls and $g$ green balls, suppose that we uniformly sample $n$ balls from the bin without replacement. Set up an appropriate martingale and use it to show that the number of red balls in the sample is tightly concentrated around $n r /(r+g)$.

Key Concepts

Azuma's Inequality

Azuma's inequality is a powerful concentration inequality that applies to martingales with bounded differences. It provides exponentially decreasing bounds on the probability that the martingale deviates significantly from its expected value, thereby serving as a critical tool in demonstrating that a random variable (or a function of the sampling process) is tightly concentrated around its mean.

Sampling Without Replacement

Sampling without replacement involves selecting items from a set where each selected item is not returned, leading to dependent selections. This situation is typically modeled by the hypergeometric distribution. The dependency structure introduced by this sampling method necessitates methods, such as martingale constructions and associated concentration inequalities, to analyze the behavior of aggregate statistics like the number of red balls drawn.

Martingales

Martingales are sequences of random variables that maintain the property that the expected value of the next term, given all the previous ones, equals the current term. This concept is central in probability theory and is used to analyze processes where there is no net expected change over time, thereby allowing one to apply stopping time techniques and concentration results.

Doob Martingale

A Doob martingale is a specific construction of a martingale associated with a given function of a set of random variables. By revealing parts of the sample sequentially, the Doob martingale provides a framework to analyze how the expected value evolves as more information is exposed, making it particularly useful in settings where one wants to control deviations from the mean.

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