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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Sigularity of the Covariance Matrix and Absolute Continuity of the Distribution [duplicate]

Suppose that X is a multivariate normal vector with the covariance matrix $\Omega$. As we know, if $\Omega$ is singular, the density of X is not defined in the usual way (because the denominator is ...
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Mathematical knowledge needed for learning upper level statistics

im a math student who wants to study statistics in depth for a master degree(maybe doctor degree). And I was wondering what kind of math knowledge might be needed for upper level statistics. I wanna ...
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If $F_X, F_Y$ agree for all $x \in \mathbb{R}$, Do their distributions $\mu_X, \mu_Y$ agree on $\mathcal{B}$?

Let random variables X and Y be defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume their distribution functions $F_X$ and $F_Y$ agree for all $x \in \mathbb{R}$. How ...
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Is $Y=Y(\omega) = \inf_{0 \leq t \leq 1}X_t(\omega) = 1_A(\omega)$ not measurable if $A \notin \mathcal{B}[0,1]?$

Consider the probability space $([0,1], \mathcal{B}[0,1],\lambda)$ where $\lambda$ is the Lebesgue measure. Let $A \subset [0,1] $ and define $ X_t(\omega) = \begin{cases} 0 \;\; if \; \; t = \omega \...
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How should we interpret random vectors in the context of statistical inference?

I am reading the book "Mathematical Statistics" from Jun Shao and I got some questions. Here is the part of the book that I am struggling with "In statistical inference and decision ...
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Kolmogorov axioms consequences

Earlier today in my stochastic processes lecture, the prof mentioned that there does not exist a probability measure P(A) defined for all subsets of [0,1] which would satisfy all the 3 Kolmogorov ...
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Why does $P_0^{(n)}(\phi_n) \leq e^{-Cn}$ implies that $\sum_{n\geq 1} P_0^{(n)}(\phi_n > e^{-Cn}) < \infty$?

In the proof of Schwartz's Theorem, the author makes the following statement By Markov’s inequality, the assumption $P_0^{(n)}(\phi_n) \leq e^{-Cn}$ implies that $\sum_{n\geq 1} P_0^{(n)}(\phi_n > ...
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What is the measure-theoretic definition of the condition probability?

Let $(S, \mathcal{S}, \Pr)$ be a probability space. Let $(\mathcal{X}, \mathcal{B})$ be a standard Borel space and a measurable map $X : S \to \mathcal{X}$. Let $(\Omega, \tau)$ be a standard Borel ...
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Singular distribution via push forward

Suppose $X$ is a multivariate normal distribution in $\mathbb{R}^3$ with a covariance matrix whose rank is 2. Therefore, it is a singular distribution. Is it possible to represent it as a push forward ...
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What is the conditional probability $P(A|B)$ in measure theory?

In Schervish's Theory of Statistics (1995) and again in A Measure Theoretic Formulation of Bayes' Theorem by @ArtemMavrin, the following equation was proved in detail $$ \mu_{\Theta \mid X}(A \...
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How do we choose the sigma algebra for a specific experiment?

I am trying to understand the connect between probability and measure theory. I get some basic things about measure theory. But what I still don't get is how do we choose the sigma algebra for our ...
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Quick question about density with respect to product measure

Suppose we got 2 non indepenendent random variables $X_1:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_1,\mathcal{B}_1)$ and $X_2:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_2,\mathcal{B}_2)$ with ...
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Continuity of conditional expectations?

An elementary result in probability theory is the so-called continuity of probability. Specifically, let $E_1\supseteq E_2\supseteq\cdots$ be a sequence of nested events. Then $P(\cap_n E_n) = \lim_{n\...
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How to build a Poisson random measure according a Levy Process

Let $(\Omega, \mathcal{F}, P)$ and $(\Theta, \mathcal{B}, \rho)$. A Poisson random measure (PRM) with intensity $\rho$ is a kernel $\mathcal{N}: \Omega \times \mathcal{B} \to \mathbb{R}$ such that: $\...
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Correct conditional expectation with respect to a different measurable space

Suppose we've got random variables $X_1:(\Omega_1,\mathcal{A}_1)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R})),X_2:(\Omega_2,\mathcal{A}_2)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ on a ...
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integral related to a general bivariate copula C(u,v) of |u-v|

I'm trying to compute the following integral over the unit square $I^2=[0,1]^2$: $$ \int_0^1\int_0^1 |u-v|dC(u,v), $$ where $C(u,v)$ is a generic bivariate copula, which should be equal to $$ 1-2\...
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Strict and Real example of Random variable and distribution [duplicate]

Now I'm studying elementary probability thoery. And It changes every abstract notions to strict text. I mean, when I was in middle school, the probability of two times coin toss is like below. $$P(HH) ...
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An extremely basic question about statistics

We do statistics on data which we assume to be outcome of random trials. The question however, is -- are we assuming that the outcome can be represented as a random variable? The outcome of an ...
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Precise Definition of $\mathbb{E}[X\mid \sigma(A)]$ Conditional Expectation of Random Variable given Sigma Algebra generated by a set

I want to define precisely, exhaustively and constructively the conditional expectation of a random variable given the sigma algebra generated by a set. This question has some discussion on it but the ...
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Conditional Expectation of Random Variable given an event

Suppose $(\Omega, \mathcal{H}, \mathbb{P})$ is a probability space, $(\mathsf{E}, \mathcal{E})$ a measurable space and $X:\Omega\to \mathsf{E}$ a random variable with well-defined expectation $\mathbb{...
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Product of kernels vs Composition of kernels

According to Wikipedia there are two main operations between two kernels: product and composition. They look almost identical to me and I cannot figure out what's the intuition between these different ...
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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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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 ...
3 votes
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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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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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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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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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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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Lebesgue-Stieltjes integration by parts on a half-open interval [duplicate]

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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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, \...
5 votes
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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 ...
4 votes
1 answer
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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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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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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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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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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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