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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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Marginalisation with respect to arbitrary distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $X: \Omega\rightarrow\mathcal{X}$ and $Y: \Omega\rightarrow\mathcal{Y}$ be random variables. Now, I know that $$\int_{C}\mathbb{P}_{...
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Distribution of the random variable $\mathbb{P}(Y|X)$

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X:(\Omega, \mathcal{A})\rightarrow(\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow(\mathcal{Y}, \mathcal{G})$ ...
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Pushforward measure for Radon Nikodym equation

Consider the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and another probability measure $\mu$, on that same space, given by $$\mu(A)=\int_A f(\omega) \mathbb{P}(d\omega)$$ Now let $X:\...
guest1's user avatar
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References to learn « Real and Functional Analysis for Statisticians »

I’m looking for a kind of book which gives the elements of Real Analysis for Mathematical Statistics and Econometric Theory, etc. I’m planning to work with the 3 following textbooks on advanced ...
Hiba__Nouhoum__Djeneba's user avatar
11 votes
3 answers
420 views

Expectation conditional on a sigma algebra, what expectation does it refer to?

In a question like Intuition for Conditional Expectation of $\sigma$-algebra a concept like $E[X|\sigma(Y)]$ is used and I am puzzled about what sort of variable this actually is. Say we have a ...
Sextus Empiricus's user avatar
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How to make sense of a uniform distribution over the real numbers (or on some other unbounded set)? [duplicate]

"Pick a random real number," seems innocuous enough. Thinking about the math though, it does not seem to work. Such a CDF would have to have a constant slope yet have $\underset{x\rightarrow ...
Dave's user avatar
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3 votes
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Can MCMC sample any probability distributions?

I have three fundamental questions related to MCMC. I would appreciate the help on any one of those. The most fundamental question in MCMC field, which I can't find a reference, is: Can MCMC generate ...
George Lu's user avatar
2 votes
1 answer
42 views

Show that the total variation distance of the Metropolis kernel to its proposal kernel is equal to the rejection probability

Furthermore, let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $Q$ be a Markov kernel on $(E,\mathcal E)$ with density $q$ with respect to $\lambda$; $\mu$ be a probability measure on $...
0xbadf00d's user avatar
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To what extent can likelihood methods be used for functional responses?

Let's suppose that we are working with a functional data set, $Y_i(t)$, $Y_i\in L^2[0,1]$, $1\le i\le n$. If we were working with univariate or even multivariate data set, likelihood methods would ...
cgmil's user avatar
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Exchanging integrals with inner products with kernel mean embeddings

I am doing some reading on kernel mean embeddings. In particular I am reading the survey paper by Muandet et al. On page 27 (Section 3.1) the authors begin a gentle introduction to kernel mean ...
Nick Bishop's user avatar
2 votes
1 answer
164 views

Questions about the conditional Radon-Nikodym derivative

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ ...
guest1's user avatar
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1 vote
1 answer
180 views

Measure Theoretic Explanation of Conditional Probability Given a Random Variable and Event

What does it mean in a rigorous measure theoretic sense to have the conditional probability of an event given a continuous random variable (or vector) and an event? As in, suppose $A,B$ are events and ...
PerpetuallyConfused's user avatar
1 vote
0 answers
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Measure Theoretic Justification For Manipulating Conditional Probability of Events Given a Continuous Random Variable and an Event [closed]

When considering the conditional probability of an event given a continuous random variable (or vector) can you essentially just manipulate the probability as if you were only working with discrete ...
PerpetuallyConfused's user avatar
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1 answer
102 views

"Almost surely" used in an expectation

Let $(\mathsf{X}, \mathcal{X})$ be a measurable space, $\pi(dx)$ be a probability measure on it, and $K:X\times\mathcal{X}\to[0, 1]$ be a Markov kernel. I have the following property $$ \int K(x, A) \...
Physics_Student's user avatar
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What is the relationship between convergence in measure and convergence in Kolmogorove distance?

Suppose I have a sequence of distribution $F_1, ..., F_n$ convergence to a limiting distribution $F$ in the sense that \begin{align} \sup ||F_n - F|| \overset{a.s.}{\rightarrow} 0 \end{align} which ...
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When running an A\B test, do we say we experiment yields two sample spaces, one for each arm, or a single sample space?

Say we have a well setup A\B test that puts half of users in one arm, half in another. Do we describe the sample space as $\Omega_1$ for arm1 and $\Omega_2$ for arm2? Or do we instead just say the ...
Estimate the estimators's user avatar
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0 answers
54 views

Calculate the cosine similarity between two probability measures, rather then between two vectors

Introduction. The cosine similarity is often calculated among two vectors, i.e. $C_S(X,Y)$: \begin{equation} C_S(X,Y) = \frac{\sum_{i=1}^{n}(X_i,Y_i)}{\sqrt{\sum_{i=1}^{n}X_{i}^{2} \sum_{i=1}^{n}Y_{i}^...
Ommo's user avatar
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1 answer
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Correspondence between the "density function of a probability measure" and the "probability density function" (PDF)

Question. If there is a one to one correspondence between a "borel probability measure" on the line $\mathbb{R}$ and a "cumulative distribution function" (CDF) (please see on page ...
Ommo's user avatar
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1 answer
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Could it make sense to calculate the "distance correlation" among two probability measures (probability distributions)?

Introduction. I post here the same question that I asked on math.stackexchange, since it might be more relevant to this forum: Could it make sense to calculate the "distance correlation" ...
Ommo's user avatar
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0 answers
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Equivalent Formulations of Thompson Sampling

I am studying Chapter 36 Thompson Sampling of the book Bandit Algorithms by Lattimore and Szepesvari. The authors present two equivalent formulations of Thompson Sampling on page 460, and I am having ...
Extrava's user avatar
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Why is this integral not Lebesgue measurable? (and how to rectify)

Suppose I have the Bernoulli random variables $S_t$ for $t=1,\cdots, T$ and for an issue related to my problem, I want to express $P[S_t=s_t\mid S_{t-1} = s_{t-1}]$, by using the law of total ...
Carl's user avatar
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1 vote
0 answers
113 views

Definition of exponent measure (extreme value theory)

Let $F$ be a distribution function on $\mathbb{R}^2$, and let $U_i$ be the left continuous inverse of $\frac{1}{1-F_i}$, where $F_i$ is the marginal distribution of $F$. In my textbook, there is the ...
Phil's user avatar
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8 votes
1 answer
352 views

Does a constant Radon-Nikodym derivative imply the measures are multiples of each other?

Let $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\pi$ and $\mu$ be two probability measures on it such that $\pi \ll \mu$ with constant Radon-Nikodym derivative $$ \frac{d\pi}{d\mu} = \text{...
Physics_Student's user avatar
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0 answers
35 views

Radon-Nikodym derivative between $\eta$ and its empirical approximation obtained through importance resampling

Suppose I have a probability measure $\eta$ on a measurable space $(\mathsf{X}, \mathcal{X})$ and I have an empirical approximation of it $$ \eta^N(dx) = \frac{1}{N}\sum_{i=1}^N \delta_{X^{(i)}}(dx) $$...
Physics_Student's user avatar
1 vote
0 answers
69 views

what is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure? [closed]

The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $...
fromscratch's user avatar
2 votes
0 answers
71 views

$1-F$ is rapidly varying if and only if there exists $b_n$ such that $\frac{\max X_i}{b_n} \to 1$ in probability

The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick. We will say $1-F$ is rapidly varying as $x \to \infty$ if $\lim_{t \to \infty} \frac{1-F(tx)}{1-F(t)} =...
Phil's user avatar
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1 vote
1 answer
75 views

Is this also the variation or extension of Continuous mapping theorem?

The common CMT (continuous mapping theorem) claim that: If $X_n \overset{p}{\rightarrow} X$, and $g$ is some continuous function, then \begin{align} g(X_n) \overset{p}{\rightarrow} g(X) \end{align} ...
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2 votes
0 answers
24 views

Change of variable formula and probability distributions [duplicate]

I think I have a misconception that I hope someone can help me clarify, I do not have a background in measure theory so that might be the problem. Say we have two random variables $X$ and $Y$ ...
Maths's user avatar
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1 vote
0 answers
36 views

Implementing Lebesgue's Dominated Convergence Theorem [closed]

I have to solve this limit using the Dominated Convergence theorem ( https://www.math3ma.com/blog/dominated-convergence-theorem ). I'm not sure what to do with that "n" before the integral. ...
user390027's user avatar
3 votes
0 answers
187 views

Minimal sufficient statistic: a measurability issue in a well-known theorem

Given a statistical model $\{\mathbb{P}_\theta\,|\,\theta\in\Theta\}$ on $(\Omega,\mathscr{F})$, and given a real-valued random variable $X$, we say a real-valued random variable $T=T(X)$ is a ...
No-one's user avatar
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1 vote
1 answer
64 views

How to prove $\int_{\mathbb{R}} g(x) dF^n(x) = n \int_{\mathbb{R}} g(x) F^{n-1}(x) dF(x)$

Let $F$ be a distribution function, and let $g \colon \mathbb{R} \to \mathbb{R}$ be a real function. I want to prove $\int_{\mathbb{R}} g(x) dF^n(x) = n \int_{\mathbb{R}} g(x) F^{n-1}(x) dF(x)$, ...
Phil's user avatar
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1 vote
0 answers
56 views

First order stochastic ordering implies countably additive probability measure?

Let $P$ be a finitely additive probability measure. I learn from my friend that: $[P(X>Y)=1 \implies \mathbb E_P[X]>\mathbb E_P[Y]]\iff$ $P$ is countably additive. Seems to be a very useful ...
dodo's user avatar
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1 vote
0 answers
16 views

Expression for Markov Kernel sampling indeces in $\{0, \ldots, T\}$ according to weights depending on another variable

I have a vector $x = (x_0, \ldots, x_T)$ and given this vector, I would like to sample an index $k$ between $0$ and $T$. The probability of sampling index $k$ is given by a weight $w_k$ that is a ...
Euler_Salter's user avatar
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3 votes
1 answer
135 views

$\limsup_n \dfrac{X_n}{n} = 0$ if $\mathbb{E}(X_1) < \infty$?

Here is an exercise in the book of author Achim Klenke. Let $(X_n)$ be iid non-negative random variables. By using Borel-Cantelli lemma, show that: $$ \limsup_n \dfrac{X_n}{n} = 0 \text{ a.s} $...
Thành Nguyễn's user avatar
3 votes
2 answers
80 views

Does strict monotonicity imply image variable is dependent with domain variable?

Suppose a random variable $X$ and a strictly monotone (and measurable) function $f$ in which $f(X)$ is always defined. Does it necessarily hold that $f(X)$ is statistically dependent with X? I should ...
Galen's user avatar
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1 vote
1 answer
109 views

Finite additive probability defined on a "finite-additive" field

Countably additive probability is defined on sigma field. However a finite additive probability needs only a "finite-additive" field: the finite additive probability does not need the ...
High GPA's user avatar
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1 vote
1 answer
191 views

Confirmation of MGF of Shifted Exponential Distribution

Consider the probability distribution $D \sim X - a$ where $X \sim \text{Exp}(\lambda)$. My task is to find the MGF of probability distribution $D$. I think I have a solution but it contradicts what I ...
YessuhYessuhYessuh's user avatar
3 votes
2 answers
880 views

Two Different Proofs of Continuous Mapping Theorem

The theorem to prove is that if $X_n$ converges weakly to $X$, and $P(X \in D_g) = 0$ where $D_g$ is the set of discontinuity of $g$, then $g(X_n)$ converges weakly to $g(X)$. In Durrett, this is ...
Phil's user avatar
  • 626
0 votes
1 answer
77 views

Inverting a characteristic function if the integral of the modulus of the cf is infinity

I'm reading a lecture slide that starts by asking if there's a way to invert a characteristic function $\psi_X$ if $\int|\psi_X(t)|~\mathrm{d}t = \infty$. From my reading, the slide then provides a ...
johnsmith's user avatar
  • 345
1 vote
1 answer
109 views

How do we define the pdf in the multi-variate case and compute expectations?

Apologies if this is a very simple question but trying to work through a result in a paper made me realize I missed something a bit fundamental in my undergrad probability and analysis courses. Lets ...
naveace's user avatar
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4 votes
2 answers
377 views

Expectations as integrals with respect to a joint distribution

As part of self-study, I'm trying to learn about conditional expectation. Example 5 here presents the problem: "Let $X_1, X_2$ be independent with $U(0, \theta)$ distribution for some known $\...
johnsmith's user avatar
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4 votes
1 answer
263 views

How to think of a sub-sigma algebra as a collection of random variables?

Where $\mathcal{H}$ is a $\sigma$-algebra on $\Omega$, section 9.1 here discusses thinking of a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{H}$ "as the collection of all numerical random ...
johnsmith's user avatar
  • 345
1 vote
2 answers
347 views

Proof of inequality $P(X \geq \lambda) \leq \frac{\sigma^2}{\sigma^2 + \lambda^2}$

Hello I have the assumptions that $E(X) = 0$ and $\operatorname{Var}(X) = {\sigma}^{2}$ . Why does this inequality hold $P(X \geq \lambda) \leq \frac{{\sigma}^{2}}{{\sigma}^{2} + {\lambda}^{2}}$ for $ ...
Jan's user avatar
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1 vote
1 answer
36 views

Equivalence of Tightness of Seqeuence of CDFs

In Durrett, a sequence of cdf $\{F_n\}$ is called tight if for all $\epsilon > 0$< there exists $M_\epsilon$ such that $\limsup 1-F_n(M_\epsilon)+F_n(-M_\epsilon) \leq \epsilon$. In Rosenthal, a ...
Phil's user avatar
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1 vote
0 answers
23 views

Prove that smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$ [duplicate]

Let $A$ be a Borel set of $\mathbb{R}$. Then show that the smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$. I am ...
Debarghya Jana's user avatar
0 votes
0 answers
88 views

Prove that smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$

Let $A$ be a Borel set of $\mathbb{R}$. Then show that the smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$. I am ...
Debarghya Jana's user avatar
1 vote
1 answer
251 views

Density from characteristic function: Durrett example 3.3.8 and 3.3.9

Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function $$f(y) = \frac{1}{2\pi} \...
Phil's user avatar
  • 626
2 votes
1 answer
116 views

Support of a continuous distribution

Suppose I have a continuous random variable $X$ on $\mathcal{R}^1$, with CDF $F(\cdot)$ and pdf $f(\cdot)$. My understanding is that there are three equivalent definitions of the support of the random ...
Ralph 's user avatar
  • 21
0 votes
0 answers
40 views

From pointwise convergence to uniform: metrics

Let $\mu_\theta$ be the limit of an empirical measure $\mu_{n, \theta}$. $\theta \in \Theta$ and $\Theta$ is a compact set. Morever, the maps $\theta \rightarrow \mu_{n, \theta}$ and $\theta \...
Eryna's user avatar
  • 309
1 vote
1 answer
206 views

Singular Distributions in Statistics

Let $X$ be a continuous random variable which induces a probability measure on $\mathbb{R}$ denoted by $\mu$. Are there any instances in statistics when we deal with random variables $X$ such that $\...

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