All Questions
Tagged with pr.probability probability-distributions
1,384 questions
1
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How to normalize an Inverse Wishart random matrix?
Background:
Let $d\in \mathbb{N}$.
Define the space of (real symmetric) positive definite matrices of size $d\times d$ as follows:
\begin{align}
\mathcal{S}_{++}^d := \big\{\mathbb{M}\in \mathbb{R}^{d\...
- 219
2
votes
1
answer
198
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Bounds for the beta CDF
This question is closely related to a previous question that I asked here:
An inequality involving the beta distribution
Let $a,b$ be strictly positive integers, and let $F_{a,b}(x)$ denote the CDF ...
- 4,049
2
votes
1
answer
167
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Hypergeometric random variables domination
Let $X\sim\text{Hypergeometric}(n,k,m)$ and $Y\sim\text{Hypergeometric}(\binom{n}{2},\binom{k}{2},M)$, where $n>k>m$ are natural numbers and $M = \binom{m}{2}$. Consider $Z = \binom{X}{2}$. I ...
- 73
1
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0
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197
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Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution
Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
- 19
6
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1
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374
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Almost evenly distributed spherical random vectors
Consider $n$ i.i.d spherically distributed random vectors $z_1 ,\cdots , z_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ ...
- 315
4
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2
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564
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A relation between the second moment of a distribution and one of its particular probability
I had recently posted a question here: To prove a relation involving a probability distribution
The relations quoted in the above question are used extensively in fluid mechanics and many other fields,...
user318428
1
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1
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216
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To prove a relation involving a probability distribution
I'm reading a book and have encountered a relation which seems to me to be impossible to prove, I would like to be sure if this is the case. The author gives a probability function as
$$p_n = \frac{e^...
user318428
4
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1
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206
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Existence of measures with given 1d marginals
This is a question about marginals of probability measures, which seems unrelated to previous questions.
Let $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ be the unit sphere. Assume that for each $\theta\in \...
2
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1
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1k
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Concentration of the norm of subGaussian random vectors
I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.
I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...
- 21
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1
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146
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Upper bound of Wasserstein distance given by subvariables of codim 1
recently I am considering the upper-bound of Wasserstain distance. Say we have random vectors $X,Y$ of dimension $n$, and let $\tilde{X}_i (\tilde{Y}_i,$ resp.) be the $(n-1)$-dim random vector of $X (...
- 363
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2
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963
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Product of three or more independent sub-Gaussian varibles
A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$.
Given a sequence of independent subgaussian ...
- 59
4
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0
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146
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An inequality for three iid random variables with a log-concave density
It was previously shown that
$$H\ge cG,\tag{1}$$
where $c:=1/14334$,
$$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$
and $X,Y,Z$ are independent random variables with the same log-concave density.
...
- 128k
0
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1
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209
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Factorisation of Gaussian random matrix into random Hermitian and correction factor
By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}_{n\times k}...
user274127
2
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1
answer
87
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Is there some similar spine decomposition for Galton-Watson tree in supercritical case whose offsprings have positive probability to have no child?
I am interested in the supercritical GW tree whose offsprings have positive probability to have no child conditioned on the event that the tree is not dead.
- 385
0
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1
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138
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Do measure-valued dynamical systems correspond to marginals of Markov processes?
Let $(\mu_n)_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}_1(X)\rightarrow \mathcal{P}_1(...
- 5,405
4
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2
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480
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Hitting probability of a line
Consider a simple (nearest neighbor) random walk on a lattice $\Bbb Z^2$ which starts at the origin, is constrained to $x\ge 0$ halfplane, and stops when it hits the line $x=n$. Denote by $p(n,k)$ ...
- 17k
3
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1
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246
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How well can we approximate a given continuous random variable by a weighted sum of several i.i.d uniform variables?
Consider a continuous random variable $X$ with the compact support $[0,1]$. For given $N\in\mathbb{N}$, we define the weighted sum as
$$
S_N=\sum_{i=1}^N a_iU_i,
$$
where $U_i$ are i.i.d. random ...
- 550
1
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0
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69
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Rate of convergence of moments
If $\mu_n$ are probability measures converging weakly to a probability measure $\mu$ and we also have convergence of even moments,$$\int x^{2k} \ d\mu
_n\rightarrow c_k+O(1/n)\ ,\ \forall k\in \mathbb{...
- 111
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0
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111
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what is the probability a moving object located inside an n by n square area gets out of the area after time t
Assume we have n by n square area and a movable object initially located at a random position in the specified area. If the object mobility modeled by a Gauss-Markov mobility model with a random speed(...
- 21
2
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1
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154
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Smooth conditional expectation with nonsmooth "reverse"
I am looking for a concrete example of the following: $(X,Y)$ are real-valued random variables such that:
$E[Y|X]$ is smooth
$E[X|Y]$ is discontinuous
Even better, I'd like to see an example where ...
- 23
0
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1
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142
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Covering number of the conditional distribution function
Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number
\begin{equation*}
\mathcal{F} = \big\{ F_{Y|W} (y | W) : y \in \mathbb{R}^d \big\}
\end{equation*}
where ...
- 331
7
votes
1
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259
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Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 128k
1
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0
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176
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Gaussian order statistics
Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one).
Suppose $X_1,\dots,X_n$ are i.i.d. standard normal.
Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
- 11
1
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1
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663
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Extreme value distribution for both minimum and maximum at the same time
I am wondering if there is an extreme value distribution that is closed under both the minimum and the maximum operation.
For example, for there is a Gumbel maximum distribution closed under the ...
- 15
4
votes
2
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683
views
Random walk on $n$-dimensional cube
Consider a symmetric random walk along the edges of an $n$-dimensional unit cube. At each time step, a particle located at a particular vertex $(a_1, \ldots, a_n)$ moves to an adjacent neighbor each ...
user171475
1
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0
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176
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Maximum mutual information of random unitary transformation
Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and identity ...
- 287
2
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1
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97
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Local limit theorems for circular/spherical distributions
Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$):
$$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/...
- 219
1
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0
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74
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Measurability of $\mathbb{R}^n$-Random Field
Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...
- 5,405
2
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0
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168
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A slight generalization of Skorokhod's representation theorem
Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random ...
- 449
5
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1
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150
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Kullback–Leibler chains
The following question was asked and then deleted by the post author:
Let $P$ and $Q$ be two probability distributions defined over the same space, with $KL(P \parallel Q) < \infty$. For $\epsilon ...
- 128k
2
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1
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230
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Mutual Information after Applying Random Unitary Matrix
Let $\mathbf{U}$ be a random unitary matrix and $\mathbf{z}$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied:
\begin{align}
\mathbf{y}=\...
- 287
1
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0
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139
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Mean-preserving spreads and equality of noise in distribution
Let $X$, $Y$ be mean preserving spreads (MPS) of the same random variable $Q$ and assume that $X =_d Y$ in distribution. Then, by the definition of MPS, there exist variables $Z$ and $Z'$ such that $Q ...
- 49
2
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1
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328
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How to check positive-definiteness of this function?
Consider a real random vector $\vec X =(X_1, X_2)$ with characteristic function $\phi(\vec t) \equiv \mathbb{E} \big[ e^{i \vec t \cdot \vec X} \big]$ (where $\vec{t}=(t_1, t_2) \in \mathbb{R}^2$) ...
- 139
5
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1
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225
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Anti-concentration of Gaussian when conditioning on event
Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
- 1,127
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133
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Convoluted Cantor-like measure which has a continuous component [duplicate]
Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable
$$
\sum_{k\ge 1}3^{-k}X_k
$$...
- 1,352
5
votes
1
answer
230
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Large deviations: Growth of empirical average of iid non-negative random varialbes with infinite expectations?
Let $X_1,X_2,X_3,...$ be iid non-negative random variables with $E[X_i]=\infty$. I am looking for references on the growth in $n$ of the empirical average under assumptions on $X_1.,..,X_n$.
A more ...
- 1,724
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1
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149
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Lottery in O(1) per participant
Goal: implement in $O(1)$ per participant a lottery where each participant has some large number of tickets, and the best (e.g. least) one wins, without actually burning electricity in proportion to ...
- 99
0
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1
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195
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Sufficient conditions for finite mean of a non-negative random variable
Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition:
$$\lim_{x\rightarrow\infty} x(1-F(x)...
- 79
2
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1
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302
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Concentration on discrete probability estimator
Let $t>1$ and $X_1,..., X_t$ a set of real random variables from a discrete distribution, whose pmf is $p(x)$, supported on the points $1,...,k$.
Let $N_t(x) = \sum_{i = 1}^t \mathbb{1}_{X_i =\, x}....
- 245
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1
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86
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Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
- 5,405
1
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0
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240
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Riemann-Stieltjes integral of a distribution function
I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a ...
- 21
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1
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235
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Existence of independent linear combinations of random variables
Let $X,Y$ be independent multivariant random variables on $\mathbb{R}^d$. Let $\alpha,\beta$ be two positive real values such that $\alpha^2+\beta^2=1$. Then, $S=\alpha X+\beta Y$ is a new random ...
- 57
1
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0
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88
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Berry-Esseen type bounds for functions of almost Gaussian random variables
Suppose that I have $n$ dependent random variables $X_1,\ldots,X_n$ with $\mathbb{E}[X_i]=0, \mathbb{E}[X_i^2]=1$, where we have the following bounds on the Kolmogorov distance from a normal ...
- 141
3
votes
2
answers
512
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Fourier transform of eigenvalue distribution of GUE matrices
I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
- 965
1
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0
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121
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Relation satisfied by a Gaussian random variable
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$:
$$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$
It seems that ...
- 171
5
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1
answer
1k
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Quantization of normal distribution
For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points.
Question: Is it known which element in $\mathcal{Q}_n$ is ...
- 1,095
0
votes
1
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478
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Covariance in the limit of random variables
Suppose $\{X_n\}$ and $\{Y_n\}$ are two sequences of random variables and we know that $X_n \overset{L^2}{\to} X$ and $Y_n \overset{L^2}{\to} Y$, where $\overset{L^2}{\to}$ means converge in mean ...
- 123
1
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1
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207
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Expectation of the sum of the squares of the cardinal of an inverse function
I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as:
$$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$
where $\oplus$ is the bitwise XOR.
...
- 167
1
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0
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233
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Maximum mutual information of a matrix representation
Let $\mathbf{Z}$ be a $m\times n$ matrix with zero-mean unit-variance i.i.d complex Gaussian entries. What is the maximum value of mutual information $I(\mathbf{X}_1,\mathbf{X}_2;\mathbf{Y})$ such ...
- 287
1
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1
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135
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KL-divergence and sub-$\sigma$-algebras
I am trying to understand if the following claim is true:
Let $P$, $Q$ be probability measures on $\mathcal{X}$. For any $\sigma$-algebra $\mathcal{G}$, with countably many atoms (sets with $\...
- 13