Math Notations

December 01, 2022 math

Notations

  • We use upper-case sans serif letters, e.g., $\mathsf{Z}$ to denote random variables with sample spaces denoted as $\mathcal{Z}$.

  • $p_{\mathsf{Z}}$ denote the distribution of $\mathsf{Z}$, which is a propability density function if $\mathsf{Z}$ is continuous. $P_{\mathsf{Z}}$ denotes the probability measure.

  • $\mathcal{P}(\mathcal{Z})$ denotes the set of all probability measures over the Borel σ-algebra on $\mathcal{Z}$.

  • $E[\mathsf{Z}]$ denotes the expectation of $\mathsf{Z}$.

  • Let $(\mathsf{Z}_1,\mathsf{Z}_2,\cdots,\mathsf{Z}_n)$ be $n$ i.i.d. copies of $\mathsf{Z}$.

  • We use $\mathsf{Z}^n$ to denote the sequence $\{\mathsf{Z}_1,\mathsf{Z}_2,\cdots,\mathsf{Z}_n\}$.

  • $|\cdot|$ denotes the Euclidean norm of a vector.

Notation for differentiation

wikipedia: https://en.wikipedia.org/wiki/Notation_for_differentiation

  • When taking the derivative of a dependent variable $y = f(x)$:
\[\frac{\dot{y}}{\dot{x}}=\dot{y}: \dot{x} \equiv \frac{d y}{d t}: \frac{d x}{d t}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{d y}{d x}=\frac{d}{d x}(f(x))=D y=f^{\prime}(x)=y^{\prime}\]

  • The value of the derivative of $y$ at a point $x = a$ may be expressed in two ways using Leibniz’s notation: \(\frac{d y}{d x}\bigg|_{x=a} \text{ or } \frac{d y}{d x}(a)\)

  • Some people also prefer useing roman type $\mathrm{d} x$ instead of italic $dx$.

  • Partial derivatives: \(\frac{\partial f}{\partial x}=f_x=\partial_x f; \frac{\partial}{\partial y}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial y \partial x}\)

  • The gradient operator can be written as \(\nabla =\bigg(\frac{\partial}{\partial y_1},\cdots,\frac{\partial}{\partial y_L}\bigg)^T\)

Big O in Probability notation

wikipedia: https://en.wikipedia.org/wiki/Big_O_in_probability_notation

$\mathsf{X}_n=o_p(a_n)$ can be written as $\frac{\mathsf{X}_n}{a_n}=o_p(1)$.

$\mathsf{X}_n=o_p(1)$ is defined as $\lim_{n\rightarrow \infty}P( \mathsf{X}_n \geq\epsilon)=0$

By $O(1)$, we denote positive absolute constants. The notation $\lfloor a\rfloor$ stands for the largest integer less than or equal to $a \in \mathbb{R}$ and $\lceil a\rceil$ for the smallest integer greater than or equal to $a \in \mathbb{R}$.

Reference

[1] List of mathematical symbols by subject

[2] Common pronunciations