All Questions
Tagged with pr.probability probability-distributions
1,384 questions
2
votes
1
answer
172
views
General definition for $k$-dependence of a family of sub-$\sigma$-algebra
If $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space, is there a general definition for the "$k$-dependence" of an arbitrary family $(\mathcal{F}_i)_{i \in I}$ of sub-$\sigma$-algebra ...
- 53
2
votes
2
answers
511
views
Geometry interpretation of any continuous random variable
Given a continuous random variable $X$ with the cdf $F_X(x)$, I want to know whether there exists a random vector $\mathbf{Z}$ uniformly distributed in a geometry region $\mathscr{Z}_n$ in $\mathbb{R}^...
- 550
2
votes
1
answer
268
views
General form for likelihood of Cox process, from Diggle–Moraga–Rowlingson–Taylor
On page 4 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the ...
- 25
0
votes
0
answers
144
views
Optimization over the set of all bounded probability measures
Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider:
$$
\max_{\mu \in \Delta^+ (X)} \theta (\mu), \...
- 67
0
votes
1
answer
141
views
Arbitrarily bad rates of convergence in Wasserstein metric
Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...
- 13
1
vote
1
answer
157
views
Constructing representations of probability revision functions
Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\...
- 631
0
votes
1
answer
268
views
Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?
From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
Tightness: Let $\Pi$ be a ...
- 657
1
vote
0
answers
68
views
(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
0
votes
1
answer
966
views
Bound the norm of sum of random vector that generated from standard basis
I have a question like this:
Consider $N$ samples $X_1, X_2, ..., X_N$ that uniformly random generated from standard basis $\{e_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\...
- 25
2
votes
1
answer
241
views
Weak continuity of law
Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
- 5,405
0
votes
1
answer
651
views
Stable law and the domains of attraction
The multivariate generalised central limit theorem with their domains of attraction was given by Rvačeva (see also this post). The original paper is not very accessible on the internet, and neither ...
- 380
2
votes
1
answer
101
views
If signed measures $\mu_n$ are such that $\mu_n\to\mu$ and $\|\mu_n\|\to c\in(0,\infty)$, does $\exp^*(\mu_n)/\|\exp^*(\mu_n)\|$ necessarily converge?
$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let
\begin{equation*}
\exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^{...
- 128k
-2
votes
1
answer
108
views
If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent
Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$.
If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
- 167
0
votes
2
answers
874
views
Bounds for the sum of dependent gaussian random variables
Let $X_1,...,X_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum_{i=1}^n w_i X_i$ be the weighted sum of these gaussian variables (because $(X_i)_{...
- 250
1
vote
1
answer
283
views
Bounding the probability Jaccard distance with total variation distance
Let $\Delta_n$ be the set of all probability vectors on $n$ points, also known as the $n$-simplex. Let $x, y \in \Delta_n$ be two probability vectors, that is, $\sum_{i=1}^n x_i = 1$ and $x_i \geq 0$, ...
- 4,466
0
votes
1
answer
294
views
Joint distribution of random Fourier coefficients
Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_{...
0
votes
1
answer
582
views
Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$
I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral
$$
\int_{\mathbb{R}^d} \log(f(x)) f(x) dx.
$$
Any references would be appreciated.
- 3
1
vote
1
answer
135
views
Probabilistic problem on the covariance matrix of a multivariate normal distribution [closed]
We have a random variable $\mathbf{X}\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with ...
- 83
7
votes
2
answers
872
views
Which random variables can be written as the difference of two independent positive random variables?
Can we characterize random variables $X$ that satisfy
$$
X\sim Y - Z
$$
for two independent positive random variables $Y$ and $Z$?
Are $Y$ and $Z$ unique in some sense?
Can (one possible choice of) $Y$...
- 302
5
votes
0
answers
130
views
Random process on a sequence of rolls of an $n$-sided die
Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a ...
- 83
1
vote
2
answers
163
views
Coupling a binomial - parity conditioning
If I have a binomial $X \sim B(n,p)$, and another binomial $X' \sim B(n,p)$ conditioned on $X'$ being of even parity. Is it true that there always exists a coupling for $(X,X')$ with $|X-X'| \le 1$? (...
- 435
2
votes
0
answers
164
views
Finding an optimal strategy for a combinatorial sequential game
We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
- 83
0
votes
1
answer
70
views
Simulation of multivariate logistic distribution conditional to a plane
For an algorithm, I have to simulate $X_1, \ldots, X_n \sim_{\text{iid}} \text{Logistic}(0,1)$ conditionally to the event $(X_1, \ldots, X_n) \in P$, where $P$ is an affine plane in $\mathbb{R}^n$.
I ...
- 2,560
0
votes
1
answer
129
views
Obtaining Chebyshev bound based thresholds for a particular tail probability using higher order moments
This is a research question. Consider an univariate non-negative random variable $q$. I intend to have a desired tail probability, say $\mathcal{A}$. If I don't know the distribution of $q$, but I ...
1
vote
1
answer
118
views
Laplace transform of the product of two gammas
Suppose that $X$ and $Y$ are both gamma distributed with shapes $a,b$ and unit scales / unit rates. To fix ideas, X has Laplace transform given by:
$$L_X(t) = \mathbb{E}(e^{-tX}) = (1 + t)^{-a}$$
How ...
- 686
1
vote
3
answers
339
views
Coupon collector targeting a collection among many
I am interested in the following problem:
We are given a universe $U$ of $n$ coupons, partitioned into $k$ collections, $C_1,\dots C_k$.
At each time step $t$, a coupon $X_t$ is selected uniformly at ...
- 111
2
votes
2
answers
206
views
non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 31
1
vote
0
answers
38
views
Grid of mostly independent variables
We're given an finite grid of random variables like so:
$$
\begin{bmatrix} A & B &... \\C & \ddots \\ \vdots&&X\\
\end{bmatrix}
$$
a subset of variables on the gird is independent ...
- 309
2
votes
3
answers
166
views
On the probability of the multivariate normal with fixed pairwise correlations being coordinate-wise non-negative
This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.
Let $(X_i:1\le i\le N)$ be a multivariate normal random vector where i) ...
- 1,096
-1
votes
1
answer
205
views
How to combine estimator with different variances?
Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$
Further, assume that we know that $Var[X_i]=\sigma_i^2$.
In the ...
- 127
0
votes
1
answer
159
views
Best bounds on the integral of an increasing function
The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a ...
- 128k
0
votes
1
answer
3k
views
In linear regression, we have 0 training error if data dimension is high, but are there similar results for other supervised learning problems?
I tried posting this question on Cross Validated (the stack exchange for statistics) but didn't get an answer, so posting here:
Let's consider a supervised learning problem where $\{(x_1,y_1) \dots (...
- 1,467
3
votes
1
answer
203
views
Underdispersed Poisson-like discrete probability distribution
I'm trying to model some discrete data that's under-dispersed enough that the Poisson distribution doesn't seem to fit. (That is, the variance is significantly less than the mean.)
If the data were ...
- 31
10
votes
4
answers
680
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 773
2
votes
0
answers
140
views
Adding elements in a list *in expectation*
Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
- 43
0
votes
1
answer
80
views
Distribution of line segment intersections in random pointsets
let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...
- 13.2k
5
votes
2
answers
174
views
Integrability of Gaussian sums
Let $(X_1, \ldots, X_n)$ be a Gaussian vector, and $Z = \sum_{i=1}^n |X_i|$.
Since the map $x \mapsto e^{x^2}$, is convex, for any $t>0$
$$
e^{tZ^2} \, = \, e^{t \big(\sum_{i=1}^n |X_i| \big)^2}...
- 51
3
votes
1
answer
630
views
Random variables with no first moment
Consider a random variable $X$ with $\mathbb E(\vert X \vert)=\infty$. I am then wondering if this implies that for $X_i$ iid copies of $X$
$$\limsup_n \frac{\vert \sum_{i=1}^n X_i \vert}{n}=\infty?$$
...
- 39
2
votes
2
answers
379
views
Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
- 1,677
0
votes
1
answer
133
views
Projection onto manifold of Gaussian measures by "trunction" of moments
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 5,405
1
vote
0
answers
100
views
Cartesian product of Poisson processes
Consider $n$ smooth, compactly supported functions $\phi_1,\dots, \phi_n \in C_c^\infty(\mathbf{R})$, and generate $n$ independent Poisson spatial processes $N_1,\dots,N_n$ on $\mathbf{R}$, each with ...
- 455
0
votes
1
answer
340
views
Expectation of the ratio of two discrete random variables with combinatorial constraints
We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$.
...
- 1,677
0
votes
1
answer
124
views
Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, \infty).$
For each $ m \ge 1$, let $X_m$ and $Y_m$ be two non-negative iid random variables with the same distribution. (The distributions of $X_m$ may change with different $m$.)
**Assume that their support of ...
- 1,467
22
votes
1
answer
5k
views
Does Pinelis' inequality (1994) exist?
I am reading a paper on stochastic optimization. And in this paper, the proofs are based on the Pinelis' 1994 inequality. I read the paper by Pinelis for more information and it is with great ...
- 339
2
votes
1
answer
1k
views
Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$
Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
- 331
3
votes
0
answers
179
views
Probability terminology
This is strictly a low-level terminology question. If I have a probability space $\Omega$ and a measurable space $S$, then a random variable $X:\Omega\rightarrow S$ gives rise via pushforward to a ...
- 23k
0
votes
2
answers
369
views
If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$
If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...
- 167
0
votes
0
answers
156
views
Total variation convergence of random matrices and convergence of empirical spectral distributions
In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
- 1
2
votes
0
answers
49
views
What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?
I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any.
I'm looking at the description of a short-term position in ...
- 223
1
vote
1
answer
154
views
If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$
Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...
- 167