All Questions
Tagged with pr.probability probability-distributions
1,384 questions
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 $X$
$$\limsup_n \frac{\vert \sum_{i=1}^n X_i \vert}{n}=\infty?$$
...
- 17.2k
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 ...
CommunityBot
- 1
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 ...
- 63.9k
0
votes
1
answer
135
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
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 63.9k
0
votes
1
answer
340
views
Expectation of the ratio of two discrete random variables with combinatorial constraints
We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$.
...
- 1,677
2
votes
1
answer
1k
views
Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$
Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
- 331
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 $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...
- 17.2k
0
votes
0
answers
156
views
Total variation convergence of random matrices and convergence of empirical spectral distributions
In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
- 1
6
votes
1
answer
3k
views
expected value of multiplication of matrices
I start with background and then ask my question, background is a brief description of wishart distribution.
Background
The Wishart distribution with $\nu$ degrees of freedom and positive definite $...
2
votes
0
answers
49
views
What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?
I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any.
I'm looking at the description of a short-term position in ...
- 223
4
votes
1
answer
134
views
The mean value of the reconstruction complexity of a random sequence
This problem is motivated by the problem of reconstructing a genome from the family of its short subwords.
Given a word $w$ and a positive integer $k$, let $M_k(w)$ be the family of all subwords of ...
- 947
1
vote
1
answer
266
views
Decomposition of the sum of nonnegative random variables [closed]
Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...
- 3,937
1
vote
1
answer
337
views
Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
- 526
8
votes
1
answer
2k
views
General Fourier inversion formula (Gil-Pelaez)
Gil-Pelaez (1951) proves the Fourier inversion formula
\begin{align*}
F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\
&= \frac{1}{2} - \frac{...
- 128k
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
0
votes
1
answer
496
views
Laplace transform inversion
I have a probability distribution that is defined through it's Laplace transform by :
$$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$
Using R and the invLT package, i have a numerical ...
- 128k
0
votes
1
answer
809
views
Concentration of $\ell_2$ norm of a vector sampled from a distribution
Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm ...
- 6,853
2
votes
1
answer
284
views
A distribution such that these expectation are 'closed-form'
I am seeking a continuous distribution with real positive support for the random variable $X$ such that, for all $t \in \mathbb R_{+}$,
$$\mathbb E \left(\ln\left(1+tX\right)\right)$$ is given in a '...
- 128k
2
votes
0
answers
84
views
approximate the square of 2-norm distance between binary distributions with high probability
Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
- 128k
4
votes
1
answer
478
views
Order statistic - Rate of convergence of a p-quantile to the expectation
Fix some $k\in\mathbb N$ and some probability $p\in[0,1]$. Denote with $F_n$ the cdf of the k-th highest oder statistic (i.e. the distribution of the k-th highest draw) of $n$ draws from a uniform ...
- 43
1
vote
0
answers
158
views
Is there a name for a random variable that is the absolute value of the difference between two iid discrete uniform variables?
I'm working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title).
That is, let $I, J$ be two iid discrete ...
- 5,429
0
votes
0
answers
92
views
Linear independence of Wishart matrices
Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
- 123
1
vote
1
answer
285
views
Exponential upper bounds for sums of martingale differences
Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}...
- 65
3
votes
1
answer
241
views
Rates of convergence for empirical quantization error
I'm looking for error rates of convergence for approximating a probability measure $P$ by a discrete probability with at most $k$ supporting points.
The setup I'm looking at is the following. Let $X$...
- 3,065
0
votes
0
answers
173
views
The reason why a test is undersized?
Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that:
$$n T_n \rightarrow_d \chi_K^2$$
under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic ...
- 331
2
votes
0
answers
243
views
An inequality of KL Divergence for two different distributions passing through a same channel
Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda$...
- 287
3
votes
2
answers
517
views
CLT for Martingales
I posted this question originally in math stack exchange, but I got no answer.
(https://math.stackexchange.com/questions/2604591/clt-for-martingales)
In wikipedia, there is a version of a CLT for ...
- 14.2k
1
vote
0
answers
46
views
How to use the mixed normal distribution to construct a proper statistics?
For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct
\begin{equation*}
\Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n
\end{equation*}
for ...
- 331
1
vote
1
answer
2k
views
Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 128k
2
votes
1
answer
905
views
Diagonalizability of Gaussian random matrices
Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
- 188k
1
vote
0
answers
33
views
Condtions for a stochastic process to be locally non-factorizable
Given a stochastic process $X=(X_t)_{t\in I}$ on $\mathbb{R}^d$ with continuous sample paths supported on a prob. space $(\Omega, \mathscr{F}, \mathbb{P})$ and such that each pair $(X_s, X_t)$, with $(...
- 463
1
vote
2
answers
113
views
If a joint density factorizes on a square, does this imply that the marginal random variables are locally independent?
Let $Z=(X,Y) : \Omega\rightarrow\mathbb{R}^2$ be a Borel-measurable random vector and $U\subset\mathbb{R}$ be open. Suppose that $Z$ is absolutely continuous with Lebesgue density $\zeta$.
I was ...
- 128k
2
votes
1
answer
160
views
A limit question involving Cramer's decomposition of normal random variables
I've come across the following question. Say we have two families of random variables, $X_N$ and $Y_N$, such that $\mathbb{E} X_N=\mathbb{E} Y_N=0$ and $\mathbb{E}X_N^2=1$. Now assume that:
$$|\mathbb{...
- 841
1
vote
1
answer
149
views
Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$
I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $...
- 1,467
2
votes
1
answer
266
views
A random variable whose characteristic function decreases the fastest
A random variable $X$ is "good" for $(a_0, b_0) \in (0,1)^2$ if its characteristic function $\varphi_X(t)$ satisfies the following constraints:
$\forall t : \varphi_X(t) \geq 0$.
$\varphi_X$...
- 128k
4
votes
1
answer
365
views
Reference for multivariate generalised CLT
I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$,
$$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\...
- 128k
0
votes
1
answer
428
views
First and last order statistics and their ratio for $\chi^2_{m}$ random samples
Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics
$...
- 5,429
2
votes
1
answer
1k
views
Sum of indicator functions of binomial random variables
Let $x_1, x_2,..., x_m$ be iid binomial random variables (each with a number of trials n and probability of success in each trial p). Define a list of binary indicator variables $y_1,y_2,...,y_m$ for ...
- 128k
4
votes
1
answer
1k
views
Functional derivative of differential entropy
I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
- 128k
0
votes
2
answers
174
views
Asymptotic properties of ANOVA when the number of groups goes to infinity
Suppose
$$X_{ij} = \mu_j + \varepsilon_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N_j$$
ANOVA can allow us to test whether $\mu_1 = \cdots = \mu_J$.
In traditional ANOVA, however, the number ...
- 364
0
votes
1
answer
220
views
Distributions associated with random sets and sums of random sets
Let's say you have an infinite random set $S$ of non-negative integers, and $T=S+S=\{x+y$ with $x,y\in S\}$. Let $N_S(z)$ be the number of elements of $S$ less than or equal to $z$; it is a random ...
- 3,098
4
votes
1
answer
320
views
The power of chi-square test
Under the null hypothesis, if we have
$$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$
the test statistic can be construct as:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$
...
- 555
2
votes
1
answer
181
views
Conditional entropy - solve example
Given a random variable $X$ that is uniformly distributed on $[-b,b]$ and $Y=g(X)$ with
$$g(x) = \begin{cases} 0, ~~~ x\in [-c,c] \\ x, ~~~ \text{else}\end{cases}$$
Now I want to compute the ...
- 131
1
vote
0
answers
212
views
A new notion of probability coupling
Let $X$ and $Y$ be two discrete random variables distributed according to $\mu$ and $\nu$, respectively. Consider the following optimization problems
$$\inf_{\pi\in \Pi(\mu, \nu)}\Pr(X\neq Y),$$
...
- 1,109
1
vote
2
answers
139
views
Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$
I'm trying to plot a graph for the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
- 128k
0
votes
0
answers
93
views
Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
- 407
0
votes
2
answers
246
views
Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v
I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
- 128k
2
votes
1
answer
300
views
On the speed of divergence of the converse of the Strong law of large numbers
By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X_1,X_2,\dots$ such that $\mathbb{P}(X_1 \ge 0)=1$ and $\mathbb{E}X_1= \infty$,
then I ...
- 14.2k
1
vote
0
answers
62
views
Quantitative bounds on convergence of Bayesian posterior
Let $Y$ be a random variable in $[0,1]$, and let $X_1, X_2, \ldots$ be a sequence of random variables in $[0,1]$. Suppose that the $X_i$'s are conditionally i.i.d given $Y$ ; in other words, I'd like ...
- 173