1
vote
3answers
106 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 ...
0
votes
0answers
86 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 ...
3
votes
1answer
375 views

Low degree polynomial approximation for the entropy function

Let $X$ be a discrete random variable with possible values $\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of $X$. In addition, denote $p_i=p(x_i)$. The entropy of $X$ is ...
0
votes
1answer
365 views

nonnegative series expansion of nonnegative functions

The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
2
votes
1answer
652 views

Probability of system failure in a distributed network

I am trying to build a mathematical model of the availability of a file in a distributed file-system. The system works like this: a node $x$ stores a file $f$ (encoed using erasure codes) at $rb$ ...
6
votes
2answers
2k views

approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...