2
votes
0answers
31 views

Where to read about this kind of “measure of irredundancy” of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction. Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...
1
vote
2answers
146 views

Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
1
vote
0answers
105 views

Uniform Law Of Iterated Logarithm for VC classes

Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. ...
-2
votes
1answer
117 views

Forms of multivariate CLT [closed]

I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
2
votes
1answer
275 views

Is this a closed set?

Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
2
votes
1answer
149 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
4
votes
0answers
63 views

Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
6
votes
2answers
405 views

When is a space of measures a measurable space?

Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
3
votes
3answers
368 views

What is the name for a non-normalized distribution?

For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
4
votes
3answers
677 views

Kolmogorov probability axioms without non-negativity condition

What is a minimal consistent modification of probability axioms to include negative values? Is it enough to use a minimal modification of axioms obtained by formal exclusion of non-negativity ...
2
votes
1answer
468 views

What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?

For integer $n$, $1 \le n \le N$, consider the random variables $X_n = \cos[t \omega_n]$ For any fixed $N$, we can take the mean $Y_N = \frac{1}{N} \sum_{n=1}^N X_n$ and define a (cumulative) ...
0
votes
1answer
910 views

basic measure theory question - measure on the natural numbers [closed]

I am looking for a succinct way to describe a subset of the natural numbers which has ``measure zero" in the following sense: Let X_1 \subset X_2 \subset .... be any strictly nesting sequence of ...
2
votes
0answers
535 views

For what sub-$\sigma$-algebra are these two measures equivalent?

In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
9
votes
2answers
559 views

Small crown probabilities (and infinite dimensional margin assumption)

My question is: How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two. Notations and definitions (to make the question rigorous) ...
18
votes
1answer
4k views

L1 distance between gaussian measures

L1 distance between gaussian measures: Definition Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
2
votes
2answers
3k views

Difference between Beta Process and Dirichlet process

I'm trying to understand the definition of a Beta process, as given in the paper: www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf The problem is that from the definition it follows that every ...
7
votes
2answers
512 views

What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?

The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...