Frequently Used Notation

$ x \stackrel{\mathrm{def}}{=}\ldots$ $ x$ is defined as ...
$ \mathbb{R}$ real numbers
$ \overline{\mathbb{R}} \stackrel{\mathrm{def}}{=}\mathbb{R}\cup \{\infty, \infty\}$
$ A^\top $ transpose of matrix $ A$
$ X \sim D$ the random variable $ X$ has distribution $ D$
$ \mathop{\text{\rm\sf E}}[X]$ expected value of random variable $ X$
$ \mathop{\mathit{Var}}(X)$ variance of random variable $ X$
$ \mathop{\hbox{Cov}}(X,Y)$ covariance of two random variables $ X$ and $ Y$
N$ (\mu, \Sigma)$ normal distribution with expectation $ \mu$ and covariance matrix $ \Sigma$, a similar notation is used if $ \Sigma$ is the correlation matrix
$ \P[A]$ or $ \P(A)$ probability of a set $ A$
$ \boldsymbol{1}$ indicator function
$ (F \circ G)(x) \stackrel{\mathrm{def}}{=}F\{G(x)\}$ for functions $ F$ and $ G$
$ x\approx y$ $ x$ is approximately equal to $ y$
$ \alpha_n = {\mathcal{O}}(\beta_n)$ iff $ \frac{\alpha_n}{\beta_n} \longrightarrow $ constant, as $ n \longrightarrow \infty$
$ \alpha_n = {\scriptstyle \mathcal{O}}(\beta_n)$ iff $ \frac{\alpha_n}{\beta_n} \longrightarrow 0$, as $ n \longrightarrow \infty$
$ {\cal F}_t$ is the information set generated by all information available at time $ t$
Let $ A_n$ and $ B_n$ be sequences of random variables.
$ A_n = {\mathcal{O}}_p(B_n)$ iff $ \forall \varepsilon > 0 \; \exists M, \; \exists N$ such that $ \P[\vert A_n/B_n\vert > M] < \varepsilon, \; \forall n>N$.
$ A_n = {\scriptstyle \mathcal{O}}_p(B_n)$ iff $ \forall \varepsilon > 0 \; : \;
\lim_{n \rightarrow \infty} \P[\vert A_n/B_n\vert > \varepsilon] = 0$.