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.
277
questions
0
votes
1
answer
18
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 ...
2
votes
2
answers
242
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 ...
- 514
0
votes
1
answer
56
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 ...
- 289
0
votes
1
answer
59
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 ...
- 55
1
vote
0
answers
26
views
Interchange of integral and differentiation [migrated]
Say I have 2 continuous Bivariate distributions( for example bivariate log normal and bivariate weibull) which have joint density functions as: $f_1(x,y,\theta)$ and $f_2(x,y,\alpha).$
Now I want to ...
- 21
4
votes
2
answers
289
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 $\...
- 289
3
votes
1
answer
83
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 ...
- 289
1
vote
2
answers
110
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 $ ...
- 35
1
vote
1
answer
28
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 ...
- 514
0
votes
0
answers
33
views
What does it mean to condition on a sub-sigma-field? [duplicate]
I am studying measure-theoretic probability independently. An earlier course on probability dealt with conditioning on an event and on a random variable.
These lecture notes discuss $P(A|C)$ where $C$ ...
- 289
1
vote
0
answers
22
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 ...
0
votes
0
answers
61
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 ...
0
votes
1
answer
73
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} \...
- 514
2
votes
1
answer
50
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 ...
- 21
0
votes
0
answers
34
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 \...
- 125
0
votes
0
answers
46
views
Intuition about conditional probability given a $\sigma$-algebra [duplicate]
I've been studying some statistics by the ways of measure theory, and came up with a problem in understanding conditional probability. The book gives the following definitions:
Let $(\Omega,\mathcal{...
- 243
2
votes
0
answers
35
views
How to (dis)prove $\lim_{k\to\infty}\lim_{n\to\infty}E(Y_{n,K}) = E(\min(X_n, K))$?
Here is the problem:
Given $X, X_1, X_2, \ldots$, non-negative random variable with finite expectation and $X_n \to X$ pointwise and $Y_{n,K} = \min(X_n,K)$, we are asked to see if
a) $\lim_{K \to \...
- 514
0
votes
0
answers
23
views
Interpretation of the inner product and projection between two density functions
Suppose I have two general density (discrete or continuous) functions $g, h \in L^2 $ defined and supported on the same domain $R\subseteq \mathbb R$.
Is the inner product $\int_R g(x) h(x) \mathrm d ...
- 2,204
0
votes
0
answers
39
views
What's the transform that relates measures on $[0,1]$ to beta distributions?
I'm looking for something that I think is called "[somebody]'s transform", but I can't remember its name.
The idea is that if I have a measure on $[0,1]$ representing the bias of an unknown ...
- 385
1
vote
0
answers
21
views
How to connect the intuitions to the math of adaptive processes?
Formal Definition
Wikipedia gives the following definition of a process adapted to a filtration:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space;
$I$ be an index set with total order $\...
- 6,135
3
votes
2
answers
146
views
Absolute Continuity of an Univariate Normal Random Variable
Consider a normally distributed random variable $X\sim N(\mu,\sigma^2)$ and a Lebesgue measure $\lambda$ on $\mathbb{R}$.
Here, the value of the distribution at $\mu$, $F_X(\mu)$, is strictly positive,...
- 755
0
votes
0
answers
29
views
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 ...
- 755
1
vote
1
answer
163
views
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 ...
- 11
2
votes
1
answer
82
views
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 ...
- 125
2
votes
1
answer
61
views
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 \...
- 125
0
votes
0
answers
15
views
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 ...
- 173
1
vote
1
answer
77
views
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 ...
- 13
2
votes
0
answers
33
views
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 > ...
- 83
3
votes
0
answers
58
views
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 ...
- 189
1
vote
0
answers
48
views
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 ...
- 81
4
votes
0
answers
76
views
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 \...
- 278
2
votes
0
answers
64
views
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 ...
- 171
0
votes
0
answers
54
views
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 ...
- 51
0
votes
0
answers
52
views
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\...
- 2,232
0
votes
0
answers
43
views
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:
$\...
- 1,027
0
votes
0
answers
37
views
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 ...
- 51
2
votes
0
answers
36
views
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
$$
1-2\...
- 31
0
votes
0
answers
17
views
Strict and Real example of Random variable and distribution [duplicate]
Now I'm studying elementary probability thoery.
And It changes every abstract notions to strict text.
I mean, when I was in middle school, the probability of two times coin toss is like below.
$$P(HH) ...
- 101
0
votes
0
answers
77
views
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 ...
1
vote
0
answers
91
views
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 ...
- 2,004
0
votes
0
answers
56
views
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{...
- 2,004
1
vote
0
answers
47
views
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 ...
- 2,004
1
vote
0
answers
36
views
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 $\...
- 61
1
vote
0
answers
22
views
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 ...
- 234
2
votes
1
answer
224
views
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 ...
- 41
2
votes
1
answer
314
views
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 ...
- 503
15
votes
7
answers
3k
views
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/...
- 7,028
1
vote
0
answers
23
views
$L^2$ 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}...
- 141
9
votes
3
answers
1k
views
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 ...
- 1,131
0
votes
1
answer
172
views
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 ...
- 2,280