본 내용은 아래의 노트 (Lecture notes in empirical process theory, Kengo Kato, 2019)를 참조하여 작성하였습니다.
empirical_process_v3.pdf
drive.google.com
- A sample space $\Omega$ is the set of all possible outcomes of a random element. A collection of subsets of a sample space $\Omega$ denoted by $\mathcal{A}.$
- A collection $\mathcal{A}$ is $\sigma$-field (or $\sigma$-algebra) if and only if
- $\emptyset \in \mathcal{A}.$ ($\Leftrightarrow \Omega \in \mathcal{A}.$)
- If a set $B \in \mathcal{A}$, then $B^c \in \mathcal{A}.$
- If a set $B_i \in \mathcal{A}, i = 1, 2, \cdots,$ then $\cup^\infty_{i = 1} B_i \in \mathcal{A}.$
- A measurable space is a double $(\Omega, \mathcal{A})$.
- Let $(\Omega, \mathcal{A})$ be a measurable space. A function $\nu$ defined on $\mathcal{A}$ is called a measure if and only if
- $0 \leq \nu(B), \forall B \in \mathcal{A}.$
- $\nu(\emptyset) = 0$
- If $B_i \in \mathcal{A}, i = 1, 2, \cdots,$ and $B_i$'s are disjoint, then $\nu(\cup B_i) = \sum \nu(B_i)$
- A measure space is a triple $(\Omega, \mathcal{A}, \nu)$.
- Let $(\Omega, \mathcal{A}, \nu)$ be a measure space. If $\nu(\Omega) = 1$, then the function $\nu$ is called a probability measure and the triple $(\Omega, \mathcal{A}, \nu)$ is called a probability space.
- Let $(\Omega, \mathcal{A}, \mathbb{P})$ be an underlying (complete, if necessary) probability space.
- For any set $T$, let $l^\infty(T)$ denote the space of all bounded functions $T \to \mathbb{R}$, equipped with the uniform norm $ \| f \| _T := \text{sup}_{t \in T} | f(t) |$.
- For a given non-empty set $T$, a non-negative function $d : T \times T \to \mathbb{R}_+$ is call a semi-metric (or pseudo-metric) if it satisfies the following three properties for all $s, t, u \in T$ :
- $d(t, t) = 0.$
- $d(s, t) = d(t, s).$ (symmetry)
- $d(s, u) \leq d(s, t) + d(t, u).$ (triangle inequality)
- If in addition $d(s, t) = 0 \Rightarrow s = t$, then $d$ is a metric. Equipped with a semi-metric $d$, $(T, d)$ is called a semi-metric space.
- For any semi-metric space $(T, d)$, let $C_u(T, d)$ denote the space of all bounded uniformly $d-$continuous functions $f : T \to \mathbb{R}$, equipped with the uniform norm $ \| \cdot \|_T$.
- If $(T, d)$ is totally bounded, then any uniformly continuous function of $T$ is bounded, and so $C_u(T, d)$ is just the set of all uniformly continuous functions on $T$.
- For any probability measure $Q$ on a measurable space $(S, \mathcal{S})$ and any measurable function $f : S \to \bar{\mathbb{R}} = [-\infty, \infty]$, we use the notation $$ Qf := \int f dQ \text{ whenever} \int f dQ \text{ exists}. $$ Further, for $1 \leq p < \infty$, let $L^p (Q)$ denote the space of all measurable functions $f : S \to \mathbb{R}$ such that $$ \| f \|_{Q,p} := (Q |f|^p)^{1/p} < \infty. $$ We also use the notation $ \| f \|_\infty := \text{sup}_{x \in S} |f(x)| $.
- The standard norm on a Euclidean space is denoted by $| \cdot |$, that is, for $a = (a_1, \cdots a_n)^\top \in \mathbb{R}^n,$ $$ |a| = \sqrt{\sum^n_{i = 1} a_i^2}. $$
- For a matrix $A$, let $\| A \|_{op}$ be the operator norm of $A$, that is, when $A$ has $d$ columns, $ \| A \|_{op} := \text{sup}_{x \in R^d, |x|=1} |Ax| $.
- For $a, b \in \mathbb{R}$, $a ∨ b = \text{max}\{a, b\}$ and $a ∧ b = \text{min}\{a, b\}$. Moreover, $a_+ = a ∨ 0$ and $a_− = (−a) ∨ 0$, so that $a = a_+ − a_−$.
- $R_+ = [0, \infty)$.
- Let $\xrightarrow{w}$ denote weak convergence, and $\xrightarrow{\mathbb{P}}$ denote convergence in probability.
- Let $(S, \mathcal{S}, P)$ be a probability space. Let $X_1, X_2, \cdots$ be i.i.d. $S$-valued random variables with common distribution $P$. We think of $X_1, X_2, \cdots ,$ when they appear, as the coordinates of the infinite product probability space $(S^\mathbb{N}, \mathcal{S}^\mathbb{N}, P^\mathbb{N})$, which may be embedded in a larger product probability space (e.g. when the symmetrization is used). For $n \in \mathbb{N}$, the empirical probability measure is defined by $$ P_n := \frac{1}{n} \sum^n_{i = 1} \delta_{X_i}. $$ For example, $$ P_n f = \int f d P_n = \frac{1}{n} \sum^n_{i = 1} f(X_i). $$
- Let $\mathcal{F}$ be a non-empty collection of measurable functions $S \to \mathbb{R}$, to which a measurable envelope $F : S \to \mathbb{R}_+$ is attached. An envelope $F$ of $\mathcal{F}$ is a function $S \to \mathbb{R}_+$ such that $$ F(x) \geq \text{sup}_{f \in F} |f(x)|, \forall x \in S.$$
- Unless otherwise stated, we at least assume that $\mathcal{F} \subset L^1(P).$ Further, to avoid measurability complications, we assume that $\mathcal{F}$ is pointwise measurable, that is, it contains a countable subset $\mathcal{G}$ such that for every $f \in \mathcal{F}$ there exists a sequence $g_m \in \mathcal{G}$ with $g_m(x) \to f(x)$ for all $x \in S$. We note here that if $\mathcal{F} \in L^1(P)$, then by the dominated convergence theorem, $\{f − P f : f \in \mathcal{F}\}$ is also pointwise measurable.
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본 내용은 아래의 노트 (Lecture notes in empirical process theory, Kengo Kato, 2019)를 참조하여 작성하였습니다.
empirical_process_v3.pdf
drive.google.com
- A sample space $\Omega$ is the set of all possible outcomes of a random element. A collection of subsets of a sample space $\Omega$ denoted by $\mathcal{A}.$
- A collection $\mathcal{A}$ is $\sigma$-field (or $\sigma$-algebra) if and only if
- $\emptyset \in \mathcal{A}.$ ($\Leftrightarrow \Omega \in \mathcal{A}.$)
- If a set $B \in \mathcal{A}$, then $B^c \in \mathcal{A}.$
- If a set $B_i \in \mathcal{A}, i = 1, 2, \cdots,$ then $\cup^\infty_{i = 1} B_i \in \mathcal{A}.$
- A measurable space is a double $(\Omega, \mathcal{A})$.
- Let $(\Omega, \mathcal{A})$ be a measurable space. A function $\nu$ defined on $\mathcal{A}$ is called a measure if and only if
- $0 \leq \nu(B), \forall B \in \mathcal{A}.$
- $\nu(\emptyset) = 0$
- If $B_i \in \mathcal{A}, i = 1, 2, \cdots,$ and $B_i$'s are disjoint, then $\nu(\cup B_i) = \sum \nu(B_i)$
- A measure space is a triple $(\Omega, \mathcal{A}, \nu)$.
- Let $(\Omega, \mathcal{A}, \nu)$ be a measure space. If $\nu(\Omega) = 1$, then the function $\nu$ is called a probability measure and the triple $(\Omega, \mathcal{A}, \nu)$ is called a probability space.
- Let $(\Omega, \mathcal{A}, \mathbb{P})$ be an underlying (complete, if necessary) probability space.
- For any set $T$, let $l^\infty(T)$ denote the space of all bounded functions $T \to \mathbb{R}$, equipped with the uniform norm $ \| f \| _T := \text{sup}_{t \in T} | f(t) |$.
- For a given non-empty set $T$, a non-negative function $d : T \times T \to \mathbb{R}_+$ is call a semi-metric (or pseudo-metric) if it satisfies the following three properties for all $s, t, u \in T$ :
- $d(t, t) = 0.$
- $d(s, t) = d(t, s).$ (symmetry)
- $d(s, u) \leq d(s, t) + d(t, u).$ (triangle inequality)
- If in addition $d(s, t) = 0 \Rightarrow s = t$, then $d$ is a metric. Equipped with a semi-metric $d$, $(T, d)$ is called a semi-metric space.
- For any semi-metric space $(T, d)$, let $C_u(T, d)$ denote the space of all bounded uniformly $d-$continuous functions $f : T \to \mathbb{R}$, equipped with the uniform norm $ \| \cdot \|_T$.
- If $(T, d)$ is totally bounded, then any uniformly continuous function of $T$ is bounded, and so $C_u(T, d)$ is just the set of all uniformly continuous functions on $T$.
- For any probability measure $Q$ on a measurable space $(S, \mathcal{S})$ and any measurable function $f : S \to \bar{\mathbb{R}} = [-\infty, \infty]$, we use the notation $$ Qf := \int f dQ \text{ whenever} \int f dQ \text{ exists}. $$ Further, for $1 \leq p < \infty$, let $L^p (Q)$ denote the space of all measurable functions $f : S \to \mathbb{R}$ such that $$ \| f \|_{Q,p} := (Q |f|^p)^{1/p} < \infty. $$ We also use the notation $ \| f \|_\infty := \text{sup}_{x \in S} |f(x)| $.
- The standard norm on a Euclidean space is denoted by $| \cdot |$, that is, for $a = (a_1, \cdots a_n)^\top \in \mathbb{R}^n,$ $$ |a| = \sqrt{\sum^n_{i = 1} a_i^2}. $$
- For a matrix $A$, let $\| A \|_{op}$ be the operator norm of $A$, that is, when $A$ has $d$ columns, $ \| A \|_{op} := \text{sup}_{x \in R^d, |x|=1} |Ax| $.
- For $a, b \in \mathbb{R}$, $a ∨ b = \text{max}\{a, b\}$ and $a ∧ b = \text{min}\{a, b\}$. Moreover, $a_+ = a ∨ 0$ and $a_− = (−a) ∨ 0$, so that $a = a_+ − a_−$.
- $R_+ = [0, \infty)$.
- Let $\xrightarrow{w}$ denote weak convergence, and $\xrightarrow{\mathbb{P}}$ denote convergence in probability.
- Let $(S, \mathcal{S}, P)$ be a probability space. Let $X_1, X_2, \cdots$ be i.i.d. $S$-valued random variables with common distribution $P$. We think of $X_1, X_2, \cdots ,$ when they appear, as the coordinates of the infinite product probability space $(S^\mathbb{N}, \mathcal{S}^\mathbb{N}, P^\mathbb{N})$, which may be embedded in a larger product probability space (e.g. when the symmetrization is used). For $n \in \mathbb{N}$, the empirical probability measure is defined by $$ P_n := \frac{1}{n} \sum^n_{i = 1} \delta_{X_i}. $$ For example, $$ P_n f = \int f d P_n = \frac{1}{n} \sum^n_{i = 1} f(X_i). $$
- Let $\mathcal{F}$ be a non-empty collection of measurable functions $S \to \mathbb{R}$, to which a measurable envelope $F : S \to \mathbb{R}_+$ is attached. An envelope $F$ of $\mathcal{F}$ is a function $S \to \mathbb{R}_+$ such that $$ F(x) \geq \text{sup}_{f \in F} |f(x)|, \forall x \in S.$$
- Unless otherwise stated, we at least assume that $\mathcal{F} \subset L^1(P).$ Further, to avoid measurability complications, we assume that $\mathcal{F}$ is pointwise measurable, that is, it contains a countable subset $\mathcal{G}$ such that for every $f \in \mathcal{F}$ there exists a sequence $g_m \in \mathcal{G}$ with $g_m(x) \to f(x)$ for all $x \in S$. We note here that if $\mathcal{F} \in L^1(P)$, then by the dominated convergence theorem, $\{f − P f : f \in \mathcal{F}\}$ is also pointwise measurable.
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