Posts tagged "math":
Characteristic Functions and Central Limit Theorem
Strong Law of Large Numbers (SLLN) and Central Limit Theorem (CLT) are two significant results in probability theory and statistics. Both theorems concern the asymptotic behavior of the sum of i.i.d. random variables, but they follow different scaling. SLLN examines the case when the sum is divided by \(n\), while CLT considers the case when the sum is divided by \(\sqrt{n}\). Their conclusions are also different. SLLN asserts that the considered random variable will converge to a constant almost surely, while CLT ensures that the distribution of the considered random variable converge to a Gaussian distribution. With the help of characteristic functions, we are able to prove the CLT straightforwardly and see how a Gaussian distribution comes out.
...Strong Law of Large Numbers
We all know that probability can be interpreted as frequency, but behind it there is an important theorem in probability and statistic theory, called Strong Law of Large Numbers (SLLN). It states that the emprical mean, i.e., the mean of samples, will converge to the expectation of the distribution almost surely. Monte Carlo integration is actually a direct application of SLLN.
...Convergence in Probability
Convergence in probability is a type of convergence of random variables on the same probability space. It is weaker than almost surely convergence but stronger than convergence in distribution.
...Quantile Functions
For a distribution function \(F:\mathbb{R}\to [0,1]\), its quatile function \(q:(0,1)\to\mathbb{R}\) is defined by1 \[ q(u):=\inf\{x:F(x)\geq u\}. \] Noting that \(F\) is right continuous, it is clear that \(\{x: F(x)\geq u\}\) is closed and bounded below for any \(0 < u < 1\). Hence, \(q(u)=\min\{x:F(x) \geq u\}\). Interestingly, there is also (see the section Properties) \[ F(x) = \sup\{u: q(u) \leq x\}.\]
...Convergence in Distribution and Weak Convergence
In statistics, we often want to study the asymptotic behavior of random variables, or in other words, the limit of their distributions. This concept appears in the central limit theorem which asserts that the empirical mean of random observations always converges to a normal distribution, regardless the distribution of the observed random variable.
...Examples of Banach Spaces
This note gives some common examples of Banach spaces as well as some counterexamples.
...Compactness
This note is a review of Chapter IV.8 of the book An introduction to set theory and topology by Freiwald.
...Optimality Conditions in Convex Optimization
Consider the following constrained optimization problem \[\begin{aligned} \min_{x\in\mathbb{R}^n}\quad & f(x) \\ \mathrm{s.t.}\quad & c_i(x) = 0,\qquad i\in\mathcal{E},\\ &c_i(x) \geq 0,\qquad i\in\mathcal{I}. \end{aligned}\] Here, \(f\) is a convex function, and \(\{c_i \mid i\in\mathcal{E}\cup\mathcal{I}\}\) are linear functions. This problem encompasses linear programming and quadratic programming, and represents a special case of general convex optimization problems. Therefore, it serves as an good starting point for learning about optimization methods.
...斗地主中的概率
问. 现有 \(N\) 个小球, 其中有 \(M\) 个作了标记. 今随机地挑出 \(K\) 个, 写出其中标记球的个数的分布.
...Helly's Selection Theorem
By Bolzano-Weierstrass’ theorem, we know that any bounded sequence of real values has a convergent subsequence. This result can be extended to finite dimensional space, i.e, any bounded sequence in \(\mathbb{R}^n\) has a convergent subsequence. However, it is not true in infinite dimensional space, say, \(\mathbb{R}^{[0,1]}\). Nevertheless, there are two well-known theorem to establish the convergence of a sequence of functions, the Arzelà–Ascoli theorem and Helly's selection theorem. The main difference between these two results is the notion of convergence of a function sequence. Arzelà–Ascoli theorem deals with the uniformly convergence and Helly's selection theorem deals with the pointwise convergence.
...Gaussian Quadrature and Borwein Integrals
We discuss how to apply the Gaussian quadrature rule to evaluate the Borwein integral \[ \textrm{pr}_h(\Delta|\bar{c}) = \frac{1}{2\pi} \int_{-\infty}^{+\infty}\cos\Delta t\prod_{m=k+1}^{k+h} \frac{\sin \bar{c}Q^mt}{\bar{c}Q^mt}\,dt, \qquad k\in\mathbb{N}, \quad h\in\mathbb{N}_+,\quad \bar{c}, \Delta\in(0,\infty). \]
...The Jordan Normal Form
我们讨论复数域 \(\mathbb{C}\) (或者任意代数闭域) 上的 \(n\) 阶方阵 \(A\) 的 对角化问题，这等价于研究线性变换 \(\mathscr{A}\colon x\mapsto Ax\) 在何 組基下有最简单的矩阵表示。代数闭的条件是为了保证我们所考虑的多项式总可 以分解为一次因式乘积。比如可以将矩阵 \(A\) 的特征多项式 \(p_\textrm{char}(x)= \operatorname{det} \,(xI-A)\) 因式分解为 \(p_\textrm{char}(x) = \prod_{j=1}^s (x-\lambda_j)^{n_j}\)，这里 \(\{\lambda_j,\; j=1, 2, \ldots, s\}\) 是 \(A\) 的特征多项式组成的集合，称 为谱，\(n_j\) 是特征值 \(\lambda_j\) 的代数重数，满足 \(\sum_{j=1}^s n_j = n\)。除了特征多项式，零化多项式也是刻画矩阵性质的有利工具。
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