All Questions
Tagged with approximation-theory pr.probability
14 questions with no upvoted or accepted answers
4
votes
0
answers
582
views
Bounds on the expectation of a function of a hypergeometric random variable: A "Jensen gap"
Main Question
Let $f:[0,1]\to [0,1]$ be continuous, let $B_n(f)$ be the $n$-th degree Bernstein polynomial of $f$, and let $r\ge 3$.
Given certain assumptions on $f$, what is an explicit and tight ...
- 697
4
votes
0
answers
269
views
Algebras and $\sigma$-algebras associated to random variables
Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^...
- 1,411
3
votes
0
answers
968
views
$\epsilon$-covering number of a set of rank-2 matrices
Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...
- 215
2
votes
0
answers
111
views
Generalization of the min-entropy that looks at the top $n$ probabilities
The min-entropy of a random variable $X$ can often be much easier to compute than the Shannon entropy. This is because the min-entropy is simply a function of the most probable value, and sometimes, ...
- 4,907
2
votes
0
answers
93
views
Approximating a probability density with a point set
Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
&...
- 4,049
2
votes
0
answers
100
views
Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes
Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
- 560
1
vote
0
answers
363
views
Incredibly accurate recursions for the Riemann Zeta function
Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here.
During some ...
- 3,098
1
vote
0
answers
103
views
Convergence result on Cornish Fisher expansion of binomial distribution
Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...
- 53
1
vote
0
answers
96
views
Relationship between Wasserstein projections and metric projections in a linear space
Let $(X,d)$ be a metric space, $x\in X$, and $Y\subset X$ is a closed set. Assume that $X$ is also a real vector space, so that we can form linear combinations over $\mathbb{R}$, but I do not assume ...
- 13
1
vote
0
answers
63
views
Approximation of measured-valued function by continuous functions
For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e.,
$$
\int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty.
$$
Let $\mu$ be a probability measure on $R^d$ such that
$$
\int_{R^d}\int_{R^d}(|z|^2\...
- 411
1
vote
0
answers
109
views
Bounding quantiles of the noncentral chi distribution
I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
- 162
0
votes
0
answers
268
views
Taylor series expansion of quantile function
Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...
- 250
0
votes
0
answers
160
views
Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
- 103
-1
votes
1
answer
129
views
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?
- 1