# 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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### 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 ...
• 1,051
106 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 ...
• 2,176
15 views

### 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 $\alpha K_1 + (1-\alpha) K_2$ ...
• 1,458
35 views

### 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 ...
• 1,682
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• 1,679
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• 873
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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 $$E\bigg[\bigg(\frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}_n^2\bigg)^{-1}\ \bigg] < \infty.$$
• 873
1 vote
273 views

• 121
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 $U$: $$h(t|U=u)=\lim_{\Delta t\rightarrow0}\frac{P(t<T<t+\Delta t|T>t,U=u)}{\Delta t}$$ ...
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 ...
• 830
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 ...
• 211
31 views

### 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 ...
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.
• 23
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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 ...
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 ...
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 ...
• 291
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\...
1 vote
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 ...
• 55
1 vote
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 $n$...