What 's the $(\Omega,\mathcal{F},P_{\theta} )$ those $T_{n}$ defined on?

Question

Definition (Consistency)

Let $T_1,T_2,\cdots,T_{n},\cdots$ be a sequence of estimators for the parameter $g(\theta)$ where $T_{n}=T_{n}(X_1,X_2,\cdots,X_{n})$ is a function of $X_{1},X_{2},\cdots,X_{n}.$

The sequence $T_{n}$ is a weakly consistent sequence of estimators for $\theta$ if for every $\varepsilon>0,$ $$\lim_{n\rightarrow\infty}P_{\theta}(|T_{n}-g(\theta)|<\varepsilon)=1.$$ If $T_{n}$ converges with probability one or almost surely (a.s.) to $g(\theta)$, that is, for every $\theta\in\Theta$ $$P_{\theta}\left(\lim_{n\rightarrow\infty}T_{n}=g(\theta)\right)=1,$$ then it is strongly consistent.

Strongly consistency implies weakly consistency.This definition says that as the sample size $n$ increases,the probability that $T_{n}$ is getting closer to $\theta$ is approaching $1$.

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I am confused about what is the ${\color{Red}{\left(\Omega,\mathcal{F},P_{\theta}\right)}}$ those $T_{n},\,n=1,2,\ldots$, defined on? What's the specific probability measure ${\color{Red} {P_{\theta}}}$ ?

Answer 1

I deem a generalized framework formalizing the concepts at work is apt here. For more details, refer to $\rm [I].$

Let $(\Omega, \boldsymbol{\mathfrak A}, \Pr)$ be a probability space. Consider the sequence of probability spaces $\langle (\mathcal X_i, \boldsymbol{\mathfrak A}_i, \mathbf P_i)\rangle_{i=1}^\infty,$ where $(\mathcal X_i, \Vert \cdot \Vert_i)$ is a normed linear space.

Consider a sequence of rvs $\langle X_i\rangle_{i=1}^\infty$ and a sequence of real numbers $\langle r_i\rangle_{i=1}^\infty.$ Then $X_n = o_P(r_n)\iff \Pr[\Vert X_n \Vert_n\leq c|r_n|] = 1, ~\forall c >0.$

Now consider a sequence of measurable functions $f_n:\mathcal X_n\to \mathcal R ,~\mathcal R$ being a normed linear space with Borel $\sigma$-field. Define $T_n := f_n(X_n)$ and $T: \Omega \to \mathcal R. $ Then $T_n$ converges in probability to $T$ if and only if $\Vert T_n - T\Vert = o_P(1).$

Now consider a parametric family of distributions $\{\mathbf P_\theta\mid \theta\in \Theta \}$ on a sequence space $\mathcal X^\infty.$ Define a measurable function $g: \Theta\to \mathcal G, ~\mathcal G$ being a metric space with Borel $\sigma$-field. Take $\mathcal X_n = \mathcal X^n$ and take measurable functions $T_n: \mathcal X_n\to \mathcal G.$ Then $T_n$ is consistent for $g(\theta)$ if for each $\theta, ~T_n\overset{\mathbf P}{\to} g(\theta).$

The simplest and most common instance is taking $\Omega = \mathbb R^\infty, ~\mathcal X_n = \mathbb R^n.$

Observe how the underlying probability space is at work here based on the implications of the characterization of the convergence in probability above.

(Also, as a footnote, one can see how in probability can be generalized: Take $S\subseteq \prod_{i=1}^\infty \mathcal X_i.$ Then $S$ occurs in probability (denoted by $\mathcal P(S)$) if, for each $i,$ there exists $S_i(\varepsilon)\in\boldsymbol{\mathfrak A}_i$ such that $\prod_{i=1}^\infty S_i(\varepsilon)\subseteq S$ and for each $\varepsilon > 0,~\mathbf P_i(S_i(\varepsilon))\geq 1-\varepsilon. $ To see how powerful it is, consider $f_n:\mathcal X_n \to \mathbb R$ and, as above, take $T_n = f_n(X_n).$ Now, define $S:= \left\{\langle x_i\rangle_{i=1}^\infty\mid\lim_{n\to\infty} f_n(x_n) = 0\right\}.$ Then $T_n = o_{\mathbf P}(1)\iff \mathcal P(S).$)

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Reference:

$\rm [I]$ Theory of Statistics, Mark J. Schervish, Springer-Verlag, $1995,$ sec. $7.1.2,$ pp. $395-398.$

Answer 2

It is customary in probability or mathematical statistics to encounter statements such as

Let $X$ be an absolutely continuous random variable with density $f$

with no reference to underlying probability space. However, we can always supply an appropriate space as follows.

Take $\Omega = \mathbb{R}$, $\mathcal{F} =$ Borel sets, $P(B) = \int_B f(x)\,dx$ for all $B\in \mathcal{F}$. If $X(w) = \omega$, $\omega \in\Omega$, then $X$ is absolutely continuous and has density $f$.

In a sense, it does not make any difference how we arrive at $\Omega$ and $P$; we may equally use a different $\Omega$ and different $P$ and a different $X$, as long as $X$ is absolutely continuous with density $f$. No matter what construction we use, we get the same essential result, that is

$$ P(X\in B) = \int_B f(x)\,dx. $$

Therefore, questions about probabilities of events involving $X$ are answered completely by knowledge of the density $f$. This implies that probabilities of events of $T(X_1,\ldots,X_n)$ are also defined by the density of $T$.

Answer 3

Let's start by setting up each individual $X_i$ as a function $X_i: \Omega_i \to S_i$, with $S_i$ being a set and $\mathcal{F}_i$ and $P_{i, \theta}$ defined appropriately. Now $T_n = T_n(X_1, X_2, ..., X_n)$ is a small abuse of notation as random variables are functions from $\Omega$, while the right hand side is a "deterministic" function $t_n$ from $S_1\times S_2\times ...\times S_n$. So I would rewrite $$ T_n := t_n(X_1, X_2, ..., X_n) $$ with $ \omega_i \in \Omega_i $ and with this we can consider $$ t_n(X_1(\omega_1), X_2(\omega_2),..., X_n(\omega_n)) = T_n(\omega_1, \omega_2, ..., \omega_n)$$ which tells us that $T_n$ is a function from $\Omega_1\times \Omega_2\times ...\times \Omega_n$, which therefore can be used to define $\Omega$. Now $\mathcal{F}$ and $P_\theta$ can be anything as long as $\mathcal{F}_i$ and $P_{i, \theta}$ are their projections down to the individual $\Omega_i$.

To answer the questions in your comments I will be a little bit more explicit: If $X_i$ are iid with $(\Omega_x, P_{x, \theta}, \mathcal{F}_x)$, then $\mathcal{F}$ is actually not $\mathcal{F}_x^n$ but instead the $\sigma$-Alegabra generated by $\mathcal{F}_x^n$ through intersection, compliments and $\sigma$-countable unions. $P_\theta$ is a probability measure that means a function from $\mathcal{F}$ to $[0, 1]$ with certain properties. Now $A \in \mathcal{F}_x^n$ then $A = A_1 \times ... \times A_n, A_i \in \mathcal{F}_x$ and $P_\theta(A) = P_{x, \theta}(A_1) \cdot ...\cdot P_{x, \theta}(A_n)$. If $A\in\mathcal{F}/\mathcal{F}_x^n $ then $P_{\theta}(A)$ is determined by how it was generated from the elements of $\mathcal{F}_x^n$

When you write $P_\theta(|T_n - \theta| < \varepsilon)$ it really means $P_\theta(\{\omega \in \Omega: |T_n(\omega) - \theta| < \varepsilon\})$, with $\{\omega \in \Omega: |T_n(\omega) - \theta| < \varepsilon\} \in \mathcal{F} $