All Questions
Tagged with pr.probability probability-distributions
1,384 questions
2
votes
0
answers
54
views
If a probability measure is a mixture of products of its marginals, does it have finite moments?
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For a linear subspace $E\subset \mathbb{R}^n$, let $\mu_E$ denote the marginal of $\mu$ on $E$. The usual orthogonal complement of $E$ is ...
- 716
1
vote
1
answer
115
views
A property of the distribution related to stochastic ordering
Let $X$ be a random variable with a symmetric support $S\subset[-M,M]$ for some $M>0$. (i.e., if x is a point of increase of CDF $F_X(\cdot)$, so is $-x$.)
Has the infimum value of $c$ such that
\...
- 128k
0
votes
0
answers
73
views
Asymptotic stochastic ordering for weighted sum of i.i.d. random variables
Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
- 19
1
vote
0
answers
68
views
Gibbs Priors form a Martingale
I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...
1
vote
1
answer
301
views
Upper-bound of the tail of a weighted sum of iid random variables
I have a question related to this one. $X_i$ are n iid random variables with CDF $1_{[0,+\infty[}(x) \Phi(x)$, i.e. it is a mixture between a folded Gaussian and a delta in $0$, both with weight $1/2$....
- 7,514
0
votes
1
answer
155
views
Limit distribution of the self-normalized sum of Cauchy random variables
This is something that has come up in my research. I originally posted this question on CrossValidated but realized it might be better suited for this site. I have deleted the question there (in case ...
- 103
1
vote
1
answer
147
views
Stochastic order on weighted sum of iid random variables
$X_i$ are n iid random variables with CDF $1_{[0,+\infty[}(x) \Phi(x)$, i.e. it is a mixture between a half Gaussian and a delta in $0$, both with weight $1/2$.
I would like to show that, $\forall a \...
- 65
0
votes
0
answers
99
views
Random walks on groups
I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
- 9
4
votes
0
answers
142
views
Algebraic area of Brownian half-plane excursion
Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path $(X_t,Y_t)...
- 3,927
7
votes
1
answer
556
views
A variation on the Borel–Cantelli lemma theme
Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let
\begin{equation*}
E:=\bigcap_{n\ge0}B_n,
\end{equation*}
where
\begin{equation*}
B_n:=\...
- 128k
1
vote
0
answers
84
views
How can one build a min-2-wise independent small sample space from min-3-wise permutations?
I have been studying a polynomial-size set of permutations from one of my lectures. The below image, taken from the lecture notes PDF, illustrates how to construct min-3-wise permutations.
My ...
- 43
1
vote
1
answer
115
views
How does Chernoff-Hoeffding bound with limited independence reduce to the usual generic CH bound with complete independence
As the title might suggest, I am referring to this paper https://www.cs.umd.edu/~srin/PDF/ch-bounds.pdf , titled : Chernoff-Hoeffding Bounds for Application with Limited Independence.
The theorem in ...
- 128k
2
votes
1
answer
138
views
expectation of the product of Gaussian kernels and their input
I was wondering if anybody knows how to solve: $$\mathbb{E}{\mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})}\left[ (\mathbf{x}{i} - \mathbf{z})(\mathbf{x}{j} - \mathbf{z})^\top \exp\left( - (\...
- 128k
0
votes
0
answers
116
views
Concentration bounds for sum of weighted sampling without replacement
Let $X$ be a collection of $2l$ non-negative numbers $X_1,X_2,\ldots,X_{2l}$. We draw $l$ weighted (proportional to values) samples without replacement from $X$. Let $S$ denote this set of $l$ samples....
- 11
14
votes
1
answer
1k
views
A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
- 3,527
-1
votes
3
answers
215
views
Proving the uniform distribution maximizes the expected value of the product of a random draw of $m$ elements from discrete distribution
Say I have a discrete probability distribution $p_i$, so $0 \le p_i \le 1$ and $\sum_i{p_i}=1$. We sample $m > 1$ draws $D$ from this distribution proportional to $p_i$ with replacement, and ...
- 1,676
1
vote
0
answers
93
views
Representation theory for symmetries of probability distribution functions
I would like to parameterize all the possible modifications to a probability density function. Is there a representation theory for this? Something along the lines of, these are all the operators $L$ ...
- 119
2
votes
1
answer
534
views
Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
- 63.9k
3
votes
1
answer
561
views
On the convergence in total variation
$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
- 4,991
3
votes
0
answers
83
views
Monotone Characteristic Function
Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$.
What other properties must $X$ have in ...
- 245
0
votes
1
answer
85
views
Conditioned on the expectation and covariance, is the total variation distance maximal for Gaussian distributions?
I want to find two distributions $p_1, p_2$, whose total variation distance is the largest between all pairs of distributions whose expectations $\mu_1, \mu_2\in \mathbb{R}^d$ and covariances $\...
- 265
1
vote
0
answers
148
views
conjecture for general form of minimax estimator
I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
- 6,541
0
votes
1
answer
69
views
Correlation for a Sum of random vectors from the sphere multiplied by matrices
Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution ...
- 3,808
1
vote
1
answer
186
views
Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
- 258
0
votes
1
answer
231
views
Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
-2
votes
1
answer
260
views
On Impossible events
Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.
Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=...
- 4,219
0
votes
0
answers
55
views
Sum of Skellam-distributed number of random variables
Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...
- 4,219
2
votes
1
answer
156
views
Some identities from graph theory and probability
The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...
- 656
4
votes
1
answer
277
views
Limit of distributions
Suppose that $X_1,X_2,\ldots, X_n$ are i.i.d random variables with continuous density $f(x)$, which is defined in the whole $\mathbb{R}$. Consider $$s(x)=\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}(\...
- 128k
2
votes
1
answer
246
views
What's the lower bound of the correlation coefficient?
Suppose a random variable $X \in \mathbb{R}$ follows a discrete distribution $p$ and takes $n$ values. We assume $E[X]=0$ and $|X|\le M$, where $M$ is a constant. Given a smooth and monotonic ...
3
votes
1
answer
2k
views
Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support
Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
CommunityBot
- 1
3
votes
0
answers
77
views
Distribution of waiting time conditioned on a fixed time length
FYI, this question is a duplicate from math stack exchange
I ask here again because I got no response.
Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
1
vote
1
answer
341
views
Form of minimax estimator
Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...
- 128k
4
votes
2
answers
2k
views
Why MLEs are asymptotically efficient whereas method of moment estimators are not?
Under appropriate regularity conditions it is well-known that Maximum Likelihood Estimation (MLE) produces asymptotically efficient estimators in the sense that their asymptotic covariance is given by ...
4
votes
2
answers
274
views
Does strong stochastic ordering exist?
For two probability measure $\mu$ and $\nu$ on $\mathbb{R}$, we call $\mu$ is stochastically smaller than $\nu$ (i.e., $\mu\leq\nu$) , if $\int f \, d\mu\leq\int f \, d\nu$ for any nonnegative bounded ...
- 2,101
1
vote
1
answer
208
views
Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance
In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\...
- 620
1
vote
0
answers
170
views
Asymptotic distribution of L infinity norm of Gaussian random vector
Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...
- 119
7
votes
0
answers
222
views
Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
1
vote
1
answer
100
views
Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?
Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let
$f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
- 128k
7
votes
2
answers
1k
views
Can a non integrable random variable satisfy a strong law of large numbers principle?
Given a random variable $X$, we denote by $X_1, X_2, \dots$ a sequence of iid copies of $X$.
Question: Does there exist a random variable $X$ with $\mathbb E[X^+] = \mathbb E[X^-] = +\infty$, but
$$\...
- 128k
4
votes
2
answers
3k
views
Entropy of the multinomial distribution
What is the entropy of the multinomial distribution? To fix notation, let us define $n > 0$ as the number of trials, $p_1, \ldots, p_k$ as the probabilities of each of the $k$ possible outcomes and ...
- 4,713
1
vote
0
answers
67
views
Random matrix theory: accounting for mean
Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{...
- 6,367
0
votes
2
answers
239
views
Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
- 37
3
votes
0
answers
125
views
Extracting moments of $\max(X_1,\ldots,X_k)$ from asymptotic behavior of $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$
For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions.
Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{...
4
votes
1
answer
518
views
Probability to return to the origin for a uniform random walk
Consider a uniform random walk on $\mathbb{R}$, with stepsize chosen uniformly from the interval $(-1,1)$. The random walk start at $x=0$. Denote by $\rho_p dx$ the probability that the random walk ...
- 128k
0
votes
1
answer
87
views
Is the $2$-point function translation invariant for general Gaussian meaures?
Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.
Next, denote a generic element of $H$ by the column ...
- 128k
0
votes
1
answer
154
views
Joint distribution of randomly permuted Poisson random variables
Let $U_1, ..., U_n$ be Poisson random variables with rates $ \lambda_1, ..., \lambda_n$ such that $\lambda =\sum_i \lambda_i = O(1)$ (i.e the sum of the rates is bounded). Suppose we have $n$ buckets. ...
- 3,937
3
votes
1
answer
453
views
When can you describe a population and its component subpopulations with the same parametric family of distributions?
I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...
- 2,821
4
votes
2
answers
403
views
What is the expected minimum total matching distance between two partitions of identically and independently distributed points?
Suppose a square $[0,1]\times [0,1]$ in which $N$ vehicles $V_i$ and $N$ riders $R_i$ are distributed identically and independently (say, uniform distribution), a bipartite matching (or a permutation, ...
- 1,352
0
votes
1
answer
112
views
Where does this coupling result use independence when bounding total variational distance?
I am reading this paper, which gives the following coupling result:
Throughout this, I'll assume the dimension $k$ is clear. Let $e_i$ be the $i$-th basis in the $k$ dimensional standard basis.
A $k$ ...
- 128k