We improve our bound from the Azuma-Hoeffding inequality for...

We improve our bound from the Azuma-Hoeffding inequality for the problem where $m$ balls are thrown into $n$ bins. We let $F$ be the number of empty bins after the $m$ balls are thrown and $X_{i}$ the bin in which the $i$ th ball lands. We define $Z_{0}=$ $\mathbf{E}[F]$ and $Z_{1}=\mathbf{E}\left[F \mid X_{1}, \ldots, X_{i}\right] .$ (a) Let $A_{i}$ denote the number of bins that are empty after the $i$ th ball is thrown. Show that in this case (b) Show that, if the $i$ th ball lands in a bin that is empty when it is thrown, then $$ Z_{i}=\left(A_{i-1}-1\right)\left(1-\frac{1}{n}\right)^{m-i} $$ (c) Show that, if the $i$ th ball lands in a bin that is not empty when it is thrown, then $$ Z_{i}=A_{i-1}\left(1-\frac{1}{n}\right)^{m-i} $$ (d) Show that the Azuma-Hoeffiding inequality of Theorem $12.6$ applies with $d_{i}=$ $(1-1 / n)^{m-i}$ (e) Using part (d), prove that $$ \operatorname{Pr}(|F-\mathbf{E}[F]| \geq \lambda) \leq 2 \mathrm{e}^{-\lambda^{2}(2 n-1) /\left(n^{2}-(\mathbf{E}(F])^{2}\right)} $$

We improve our bound from the Azuma-Hoeffding inequality for the problem where $m$ balls are thrown into $n$ bins. We let $F$ be the number of empty bins after the $m$ balls are thrown and $X_{i}$ the bin in which the $i$ th ball lands. We define $Z_{0}=$ $\mathbf{E}[F]$ and $Z_{1}=\mathbf{E}\left[F \mid X_{1}, \ldots, X_{i}\right] .$
(a) Let $A_{i}$ denote the number of bins that are empty after the $i$ th ball is thrown. Show that in this case (b) Show that, if the $i$ th ball lands in a bin that is empty when it is thrown, then
$$
Z_{i}=\left(A_{i-1}-1\right)\left(1-\frac{1}{n}\right)^{m-i}
$$
(c) Show that, if the $i$ th ball lands in a bin that is not empty when it is thrown, then
$$
Z_{i}=A_{i-1}\left(1-\frac{1}{n}\right)^{m-i}
$$
(d) Show that the Azuma-Hoeffiding inequality of Theorem $12.6$ applies with $d_{i}=$ $(1-1 / n)^{m-i}$
(e) Using part (d), prove that
$$
\operatorname{Pr}(|F-\mathbf{E}[F]| \geq \lambda) \leq 2 \mathrm{e}^{-\lambda^{2}(2 n-1) /\left(n^{2}-(\mathbf{E}(F])^{2}\right)}
$$

Key Concepts

Azuma-Hoeffding Inequality

The Azuma-Hoeffding inequality is a concentration result that provides bounds on the deviations of a martingale (or a function with bounded differences) from its mean. It is particularly useful when dealing with sums of dependent random variables, as long as the change in the sum when a single variable is adjusted is bounded. This inequality helps in establishing tail bounds, which are crucial in probabilistic analyses of randomized algorithms and combinatorial structures.

Doob Martingale

A Doob martingale is formed by taking the conditional expectations of a random variable with respect to an increasing sequence of sigma-algebras. In many probabilistic problems, including occupancy problems, the Doob martingale is used to analyze how the expected outcome evolves as more information becomes available, making it a useful tool in applying concentration inequalities like the Azuma-Hoeffding inequality.

Bounded Differences Method

The bounded differences method is a technique used to control how much a function of several random variables can change when one variable is altered. This method is directly involved in applying the Azuma-Hoeffding inequality because it requires knowledge of the maximum change (or 'difference') in the function's value due to a single change in its arguments. In the context of balls-and-bins problems, establishing these bounds is critical to deriving concentration results.

Occupancy Problems in Probability

Occupancy problems involve distributing items (such as balls) into containers (such as bins) and analyzing the resulting distribution. A common question is determining the number of empty bins after a number of balls are thrown. Such problems are classical in probability theory and combinatorics, providing valuable insights into random allocation and uniform distributions.

Conditional Expectation

Conditional expectation is the expected value of a random variable given that certain conditions or events are known. It plays a key role in constructing martingales and in formulating the step-by-step analysis required in many probabilistic proofs. In the context of the problem, conditional expectations are used to update the expected number of empty bins as each ball is thrown, which is vital for the application of the Azuma-Hoeffding inequality.

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