Questions tagged [measure-theory]

A field of mathematics concerning measures, which are functions mapping sets to real numbers that generalize the idea of area and volume. Measure theory underlies modern treatments of probability. Questions about measure theory with a weak connection to probability or statistics may be more suitable for math.stackexchange.com than this site.

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Visualization of $L(X)$

Let $L^2_+$ the set of all $2$-dimensional nonnegative random vectors $X = (X_1, X_2)^⊤$ with finite and positive marginal expectations, and let $Ψ^{(2)}$ the class of all measurable functions from $\...
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Computing certain integrals with a particular distribution function

Let $(\xi_n, n \in \mathbb{N})$ be a sequence of random vector such that $G_n$ is the distribution function of $\xi_n$. Let $\big(\psi_{j}^n \big)_{(j,n) \in \mathbb{N}^2} $ be a double sequence such ...
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How to formally define a conditional distribution conditioning on an event of probability zero?

Given $[X,Y]\sim N(0,I_2)$, a intuitive guess of the value of $P(X=x|\{X,Y\}=\{x,y\})$, where $\{\}$ means unordered set, is $1/2$ by symmetry. This type of notations is typically applied in ...
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What is the difference between a probability measure and a probability density function? [duplicate]

During my research, I have repeatedly come across the terms probability measure and probability density function (pdf). I am familiar with the concept of a pdf, but I am not entirely sure how ...
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Difference in Probability Measure vs. Probability Distribution

I am trying to better understand the Difference in "Probability Measure" and "Probability Distribution" I came across the following link : https://math.stackexchange.com/questions/...
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$L^2$ convergence of inverse

Let $h$ be some bounded non-negative function. Assume that some random quantity $\mu^N (h)$ be some random quantity with almost sure limit $\mu(h) > 0$. For instance we could have $\mu^N(h) = N^{-1}...
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3 answers
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Confidence interval / p-value duality: don't they use different distributions?

Idea: p-value is less than the level of significance if and only if the corresponding CI does not include the null value; and vica versa, the p-value is greater than the level of significance if and ...
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If every event is trivial (0 or 1 probability), then every random variable is (a.s.) degenerate/constant. Maybe Lebesgue decomposition?

There are these: 1, 2, 3, but I wanna try different ways. Let $(\Omega, \mathfrak{F}, \mathbb P)$ be such probability space with each (event) $E \in \mathfrak F$ having trivial probability. Consider a ...
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Effective Sample Size for a cycle or mixture of kernels

The Effective Sample Size (ESS) of a univariate Markov Chain can be used to assess it performance. Is there a version of the ESS for when the Markov Kernel is a mixture $\alpha K_1 + (1-\alpha) K_2$ ...
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Suppose $X$ follows normal and $Y$ follows Bernoulli, wrt which measure does $(X,Y)=(-0.005,1)$ have a measure zero?

Suppose $X$ follows standard normal distribution and $Y\in\{0,1\}$ follows Bernoulli(0.5),i.e., $Pr(Y=1)=0.5$. Intuitively, I know the point or event $(X,Y)=(-0.005,1)$ has a measure of zero. But I ...
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Convergence of an exponential martingale

Suppose that $\{Y_n : n \geq 1 \}$ is a sequence of i.i.d. random variables with common distribution $N(0,1)$. Let $\{a_n : n \geq 1 \}$ be a sequence of real numbers. Set $X_0 \equiv 1$ and for $n \...
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Using the Dominated convergence theorem in a sequence of Indicator functions

Let $Z_t\sim WN(0,\sigma^2)$ be a white noise. Consider a MA(q) processes: \begin{equation} X_t^q = \sum_{j=0}^{q} \theta_j Z_{t-j}, \quad X_t = \sum_{j=0}^{\infty} \theta_j Z_{t-j} \end{equation} ...
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Measure-theoretically rigorous treatment of statistical learning theory

My main source on statistical learning theory has been Shwartz/Ben-David. This is a good book but it's a little vague from a measure-theoretic point of view. For example, in the definition of PAC ...
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What does "$\pi$-almost" mean?

I'm reading "Complex Stochastic Systems" edited by Barndorff-Nielsen et al. I found that they use the expression $\pi$-almost consistently throughout Chapter 1. I don't understand what $\pi$-...
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Easy to follow video lectures or online help on measure-theoretic mathematical statistics

Due to some unexpected events, I was not able to follow my measure-theoretic mathematical statistics classes for a while and now I have to cover these materials (chapter 1 of Jun Shao's book and ...
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1 answer
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Proof of Weak Convergence

Suppose that we have two cumulative distributions $F_{n}(x)=\left\{\begin{matrix} 1, \ x\in [\frac{1}{n},\infty) \\ 0, \ otherwise \end{matrix}\right.$ and $F(x) =\left\{\begin{matrix} 1, \ x\in [0,\...
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Show that this probability distribution is an exponential family

We define a probability distribution on the non-negative integers 0,1,2,...,as having point probabilities $$P_\eta(k)=G(k,\rho)\left(\frac{1}{\rho+\eta}\right)^{\rho}\left(\frac{\eta}{\rho+ \eta}\...
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2 votes
1 answer
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Gaussian process with polynomial covariance function integrating to 0

In Rasmussen and Williams (2006, p. 88), one can find the following piecewise polynomial covariance function with compact support: $$ k_\text{pp}(t,t') = (1-|t-t'|)_+ $$ where $(x)_+ = x \times [x>...
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  • 155
2 votes
1 answer
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How is the asymptotic justification of the "linearization by influence function method" for surveys established?

The survey R package recently adopted the "linearization by influence function" method of estimating covariances between domain estimates. The central ...
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2 votes
1 answer
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Notation for a derived probability measure

Suppose that I have a random variable $X$, which is a variable-dimensional (i.e. it could be 1 dimensional, 2 dimensional, 3 dimensional, etc.). The dimension of a specific $x$ is known and denoted as ...
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2 votes
1 answer
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Differentiation of a distribution function with respect to a measure

The context is that $X^{(n)}=(X_1,\ldots,X_n)$ consists of $n$ $i.i.d.$ observations according to $F$. Assume $F$ is dominated by a common $\sigma$-finite measure $\mu$, and let $f=\frac{dF(x)}{d\mu}$...
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6 votes
0 answers
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Lebesgue-Stieltjes integration by parts on a half-open interval

I have run into a problem in a proof of the bound for the rate convergence of an empirical risk function based on unbounded loss to the true model risk (Vapnik, Statistical Learning Theory, Theorem 5....
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4 votes
1 answer
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When was a random variable first called a "random variable"? And why is it called as such?

From measure theoretic foundations, it is clear that a random variable is neither random nor a variable. It is a deterministic function developed as follows: Construct probability space: $(\Omega, \...
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5 votes
1 answer
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Probability of getting a rational number for continuum trials

As far as I know, the probability of getting a rational number from the interval $(0, 1)$ is zero if we follow the Lebesgue measure. Still, it doesn't mean it's impossible. Now, suppose we have a ...
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3 votes
1 answer
747 views

Regular conditional distribution vs conditional distribution

What is the difference between the concept of the regular conditional distribution and the concept of the conditional distribution? Why do we need these two different concepts? Under which ...
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1 vote
0 answers
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Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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2 votes
1 answer
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Joint probability measure

I know from my measure theory class that for two $\sigma$-finite measure spaces $(\mathcal{X}_1, \mathcal{A}_1, \mu_1)$ and $(\mathcal{X}_2, \mathcal{A}_2, \mu_2)$ there exists a unique measure $\mu :=...
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  • 587
2 votes
2 answers
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Express expectation value of a joint distribution over a discrete and continuous random variable

Let $Y$ be a discrete random variable and let $X$ be an (absolutely) continuous random variable and $f(X, Y)$ a function of these two random variables. Let $P(X, Y)$ be the joint probability measure. ...
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3 votes
1 answer
383 views

Why do we add an extra decimal place when we calculate the range in statistics?

From Mario Triola's textbook: I understand that we add a decimal place when we calculate the median (it is possible to have (a+b)/2 ). I absolutely do NOT understand why we do that when we round off ...
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4 votes
1 answer
137 views

Practical example of a non-measurable maximum likelihood estimator

This post gives an example of a situation where the MLE is not measurable. However, this doesn't seem to be a situation that you would ever encounter in practice. Is there a more practical example of ...
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1 vote
0 answers
59 views

Probability and measure theory problem

I am self learning the measure and probability theory and came across this problem. This might look very simple but appreciate any help or a hint in the right direction. When $X$ is a random variable ...
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8 votes
1 answer
507 views

Limit of Integration of continuous function

How to evaluate the following limit- $$\lim_{n \to \infty} \int_0^1 \int_0^1\cdots\int_0^1 f \bigg(\frac{x_1 + x_2 + \cdots + x_n}{n} \bigg) dx_1 dx_2....dx_n$$. Here $f()$ is a continuous function $f:...
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0 answers
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Is the inverse of the sample variance uniformly integrable?

Let $X_1,X_2,\dots,X_n$ be a sample of $n$ independent and identically distributed observations of a continuous population random variable $X$. Define $Z_n$ to be the inverse of the sample variance: $$...
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Is the inverse of the sample variance integrable?

Is the inverse of the sample variance integrable? That is, does it hold that $$ E\bigg[\bigg(\frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}_n^2\bigg)^{-1}\ \bigg] < \infty. $$
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1 vote
0 answers
273 views

Chain rule for KL divergence, conditional measures

The chain rule for KL divergence is widely seen in the theoretical machine learning literature and generally referenced to [1, Theorem 2.5.3]: $$ \text{KL}[p(x, y) \mid q(x, y)]= \text{KL}[p(x) \mid q(...
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3 votes
2 answers
32 views

Relationship between deterministc function of random variables

Given a discrete $P(X,Y,Z)$ let's call $\Omega$ the set of all deterministic functions $f: XYZ \rightarrow W$ and $\Omega'$ the set of all deterministic functions $f': XY \rightarrow V$. Is it correct ...
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2 votes
1 answer
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Show $X_n \to 0$ in probability

I am asked to show : Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define $X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...
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2 votes
1 answer
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The book Probability Theory by Klenke, Exercise 1.2.1, I wonder if it’s an error?

The book Probability Theory by Klenke has for Exercise 1.2.1 the following: Let $$A =\{(a,b]\cap \mathbb Q : \ a,b\in\mathbb R, \ a\le b\}.$$ Define $$\mu: A \to [0,\infty)$$ by $$ \mu((a,b]\cap \...
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4 votes
1 answer
104 views

Does the law of total probability apply to hazards?

Consider the hazard function for a random variable $T$, conditional on some other random variable $U$: $$ h(t|U=u)=\lim_{\Delta t\rightarrow0}\frac{P(t<T<t+\Delta t|T>t,U=u)}{\Delta t} $$ ...
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4 votes
1 answer
89 views

Is $\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ uniformly integrable (UI)? What assumptions make it UI?

$\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ Let $\mathbf{X}_n$ be the usual data matrix in standard multiple regression where I have used the subscript $n$ to indicate the number ...
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3 votes
3 answers
229 views

Is it possible to interchange the quantile operator and a measurable monotone function? $Q_\theta(f(X)) = f(Q_\theta(X))$

Let $Q_\theta(X)$ is the $\theta^{th}$ quantile of a random variable $X$, and if $f$ is a measurable strictly increasing function. I want to know if $Q_\theta(f(X)) = f(Q_\theta(X))$. I know that for ...
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0 votes
0 answers
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What's the term for a r.v. X "upper bounding" Y probabilistically? [duplicate]

What's the term for when a random variable $X$ has a higher probability of being greater than t than the probability of $Y$ being greater than $t$ for all real $t$? I recall reading a term for this ...
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2 votes
1 answer
645 views

Show that maximum of two random variables is a random variable

If we have that $X$ and $Y$ are random variables, how do we prove that $Z=max(X,Y)$ is also a random variable ? I want to do this by showing that $Z$ is measurable, but I don't know how to do this.
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2 votes
1 answer
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What is the proper (measure theoretic) definition of a random variable?

(1) If we believe the following link: http://www.columbia.edu/~md3405/DT_Risk_2_15.pdf Then a random variable is a map from a probability/measurable space to a metric space, i.e. (2) If we believe ...
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0 votes
0 answers
51 views

Intermediate proof for Glivenko-Cantelli Theorem

Show that for any cdf, the following holds: $sup_{x\in \mathbb{R}}|F_n(x)-F(x)|\leq sup_{u\in [0,1]}|\overline{F}_n(u)-F(u)|$ Where $F_n:=\frac{1}{n}\sum_{i=1}^n1_{(-\infty,x]}(X_i)$ is the empirical ...
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3 votes
1 answer
757 views

How to understand conditional expectation w.r.t sigma-algebra: is the conditional expectation unique in this example?

This board has many questions on how to understand conditional expectations w.r.t a $\sigma$-algebra; it's clearly a topic that confuses many. I am one of them. I read all the other questions and ...
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0 answers
30 views

Covariance and indicator function relations

Let $\mathcal A$ and $\mathcal B$ be two (sub) sigma-algebras of some probability space. Rio (2017), pp. 4 defines the coefficient: $$\alpha(\mathcal A,\mathcal B)=2\sup\{\lvert Cov(1_A,1_B):(A,B)\in\...
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1 vote
0 answers
379 views

Product of two probability density function

Suppose $f$ and $g$ are two probability density functions. I have seen economists use $\int f(x)g(x) dx$ as some kind of similarity measure. For example, Jaffe (1986) uses sum of product of two ...
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1 vote
1 answer
43 views

Independence of a statistic from size of sample

If you have some (non-constant) statistic $T(X_1, X_2,...,X_n) = f(X_1,X_2,...,X_n)$, is it independent of the number of elements in the sample $(X_1, X_2, ..., X_n)$? That is, is it independent of $n$...
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1 vote
0 answers
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Difference between measure and metric in context to information and mutual information

In measure theory, a measure is intrinsic property of the set which informs about the size of the set. On the otherhand an information say a mutual information is a diminishing return property, that ...
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