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Let $t_1, \ldots, t_n$ be jointly independent boolean random variables. In Corollary $1.9$ we established a large deviation inequality for the polynomial $t_1+\cdots+t_n$. In many applications, it is also of interest to obtain large deviation inequalities for more general polynomials $P\left(t_1, \ldots, t_n\right)$ of the boolean variables $t_1, \ldots, t_n$. One particularly important case is that of a boolean polynomial
$$X:=\sum_{A \in \mathcal{A}} \prod_{j \in A} t_j,$$
where $\mathcal{A}$ is some collection of non-empty subsets of $[1, n]$. Observe that boolean polynomials are automatically positive and monotone increasing, and hence any two boolean polynomials are positively correlated via the FKG inequality (Theorem 1.19). More generally, if $X$ and $Y$ are boolean polynomials, then $f(X)$ and $f(Y)$ will be positively correlated whenever $f$ is a monotone increasing or decreasing function. In particular, we see that
$$\mathbf{E}\left(e^{-s(X+Y)}\right) \geq \mathbf{E}\left(e^{-s X}\right) \mathbf{E}\left(e^{-s Y}\right)$$
for any real number $s$. Using this fact, the exponential moment method, and some additional convexity arguments, Janson [190] derived a powerful bound for the lower tail probability $\mathbf{P}(X \leq \mathbf{E}(X)-T)$ :

Theorem $1.28$ (Janson’s inequality) Let $t_1, \ldots, t_n, \mathcal{A}, X$ be as above. Then for any $0 \leq T \leq \mathbf{E}(X)$ we have the lower tail estimate
$$\mathbf{P}(X \leq E(X)-T) \leq \exp \left(-\frac{T^2}{2 \Delta}\right)$$

where
$$\Delta=\sum_{A, B \in \mathcal{A}: A \cap B \neq \emptyset} \mathbf{E}\left(\prod_{j \in A \cup B} t_j\right) .$$
In particular, we have
$$\mathbf{P}(X=0) \leq \exp \left(-\frac{\mathbf{E}(X)^2}{2 \Delta}\right)$$

In previous sections, we often considered a polynomial $Y=Y\left(t_1, \ldots, t_n\right)$ of $n$ independent random variables $t_1, \ldots, t_n$, and wished to control the tail distribution of $Y$. For instance Chernoff’s inequality shows that the polynomial $t_1+\cdots+t_n$ is concentrated around its mean, while Janson’s inequality shows that the values of certain polynomials (especially those of low degree) could very rarely be significantly less than the mean.

In this section, we present some further results of this type, that assert that certain polynomials with small degrees are strongly concentrated. These results can be seen as generalizing Chernoff’s bound, and also provide (in certain cases) the missing half (upper tail bound) of Janson’s inequality.

To motivate the results, let us first give a classical result which works for any function $Y$ (not just a polynomial) provided that the Lipschitz constant of $Y$ is small.
Lemma $1.34$ (Lipschitz concentration inequality) Let $Y:{0,1}^n \rightarrow \mathbf{R}$ be a function such that $\left|Y(t)-Y\left(t^{\prime}\right)\right| \leq K$ whenever $t, t^{\prime} \in{0,1}^n$ differ in only one coordinate. Then if $t_1, \ldots, t_n$ are independent boolean variables, we have
$$\mathbf{P}\left(\left|Y\left(t_1, \ldots, t_n\right)-\mathbf{E}\left(Y\left(t_1, \ldots, t_n\right)\right)\right| \geq \lambda K \sqrt{n}\right) \leq 2 e^{-\lambda^2 / 2}$$
for all $\lambda>0$.

## 加性组合代写

$$X:=\sum_{A \in \mathcal{A}} \prod_{j \in A} t_j,$$

(定理 1.19) 。而一般地说，如果 $X$ 和 $Y$ 是布尔多项式，那 $(X)$ 和 $f(Y)$ 将呈正相关，无论何时 $f$ 是单调迫增或递减函数。特别 是，我们看到
$$\mathbf{E}\left(e^{-s(X+Y)}\right) \geq \mathbf{E}\left(e^{-s X}\right) \mathbf{E}\left(e^{-s Y}\right)$$

$$\mathbf{P}(X \leq E(X)-T) \leq \exp \left(-\frac{T^2}{2 \Delta}\right)$$

$$\Delta=\sum_{A, B \in \mathcal{A}: A \cap B \neq \emptyset} \mathbf{E}\left(\prod_{j \in A \cup B} t_j\right)$$

$$\mathbf{P}(X=0) \leq \exp \left(-\frac{\mathbf{E}(X)^2}{2 \Delta}\right)$$

$$\mathbf{P}\left(\left|Y\left(t_1, \ldots, t_n\right)-\mathbf{E}\left(Y\left(t_1, \ldots, t_n\right)\right)\right| \geq \lambda K \sqrt{n}\right) \leq 2 e^{-\lambda^2 / 2}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。