All Questions
Tagged with pr.probability probability-distributions
1,384 questions
6
votes
1
answer
499
views
Maxima of Brownian motion
It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s..
From a physics perspective it seems reasonable that when the disorder of the path of a ...
- 536
5
votes
0
answers
711
views
Concentration inequality for max component of a multivariate Gaussian in the general case
I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
- 283
2
votes
1
answer
675
views
Moment generating function of random unit vector
Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of
$$\mathbb{E}[\exp(X^Tv)]$$
for any $v$?
- 1,495
4
votes
0
answers
261
views
Tight bounds for finite de Finetti's theorem
de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following ...
- 763
4
votes
1
answer
119
views
Expected value of a random variable conditioned on a positively correlated event
I have a random variable $x \in [a, b]$ with PDF $f(x)$ and an event $E$ which satisfies the following property for any $x'<b$.
$$\Pr[E\mid x > x'] \geq \Pr[E]$$
My question is whether or not ...
- 189
1
vote
1
answer
194
views
Is there a 1/poly(n) or 1/polylogn upper-bound for this tail bound?
Is there a good tail bound for $\operatorname{P}\!\Bigg[\bigg\vert\dfrac{\sum_{j=1}^n(\sum_{i=1}^n a_{i,j})^2}{n^2} -1\bigg\vert > \epsilon\Bigg]\,,$ where all $a_{i,j}$'s are iid, with $\...
- 75
3
votes
1
answer
226
views
Total offspring of Poisson multitype branching process
A normal branching process $Z_n$ initialized with $Z_0=1$ and offspring generated from $Pois(p),p<1,$ has a total progeny / total off spring distribution
$$X=\sum_{n=0}^\infty Z_n$$
$X\in \mathbb{...
- 315
3
votes
1
answer
456
views
Heavy tail central limit theorem
I am looking for a proof based on characteristic functions for the generalized central limit theorem when the second moment does not exist, in which case one ends up with a power law rather than a ...
5
votes
1
answer
1k
views
Sum of random variables are equal in distribution
Suppose that $X,Y$ are scalar random variables supported on some standard Lebesgue probability space $(\Omega, \mathrm{P})$, such that $X \overset{\mathrm{d}}{=} Y$ in the sense that their pushforward ...
- 85
1
vote
0
answers
72
views
Large Deviation of Triple Poisson Product
Let $X_i$ with $i=1,\ldots,n$ be independent Poisson variables, $X_i$ with parameter $\lambda_i.$
Let $\circ$ be a group operation on a group of size $n.$
I would like to obtain a large deviation ...
- 10.4k
1
vote
1
answer
134
views
Random optimization problem
Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution ...
- 1,677
4
votes
1
answer
142
views
Linear combination of coordinates of random unit vector
Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda_1,...,\lambda_n$ be given real numbers. What is the distribution of
$$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$
Does it happen ...
- 1,495
2
votes
1
answer
206
views
Density of random matrix only depends on its spectrum
Suppose a random positive definite matrix $A\in\mathbb{R}^{n\times n}$ has density function (with respect to the lebesgue measure on $\mathbb{R}^{n(n+1)/2}$) $f(A)=g(\lambda_1(A),...,\lambda_n(A))$ ...
- 1,495
1
vote
0
answers
98
views
Joint distribution of two weighted sums of IID random variables
Let $X_1, X_2, \dots$ be independently uniformly distributed random variables in $\{-1, +1\}$ and let $a_1, b_1,a_2,b_2, \ldots \in \mathbb{R}$ be fixed, bounded and of non-zero average. Let $Y_n=...
- 341
2
votes
0
answers
57
views
Given a set of marginals, what is the largest support of a distribution satisfying these?
Given a random variable $X$ with support over $\{0,1\}^I$, we can define the marginal distribution on the bits indexed by $A \subseteq I$ by $Pr(X_A = x_A) = \sum_{x \in \{0,1\}^{I - A}} Pr(X = x \cup ...
1
vote
1
answer
3k
views
Correlation between square of normal random variables
Suppose I have $X,Y$ bivariate normal with correlation coefficient $\rho \in (0,1)$ . Then , what is the correlation between $X^2 $ and $Y^2$ ?
I am aware of the fact that the square of the normal ...
- 183
5
votes
3
answers
512
views
Optimisation under constraint of Wasserstein distance
Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or ...
- 111
1
vote
1
answer
90
views
The weak version of the memoriless property
In our group we are working with a probability distribution $X$ defined on a non-negative domain, satisfying the following property
$$
P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right],
$$
...
- 396
8
votes
1
answer
355
views
Lower Bound of KL-Divergence Between Two Gibbs Measures
Suppose we have two Gibbs measures with densities
$$
p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)).
$$
Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, ...
- 1,127
2
votes
0
answers
70
views
If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...
- 167
5
votes
2
answers
185
views
Density near at $0$ for the integral of the positive part of the Brownian motion
This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
- 128k
1
vote
0
answers
103
views
A question about pdfs with likelihood ratio order
Suppose $f_1,f_2,\dots$ are pdfs of absolutely continuous random variables with the same support (say an interval). Assume that $\{f_i\}$ are strictly positive in their support. Furthermore, $\frac{...
- 393
3
votes
1
answer
3k
views
Is there a tight lower bound for the expectation of the product of two positive valued random variables?
Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
...
1
vote
1
answer
92
views
boundig variation from median [closed]
Given a scalar random variable $X$, suppose that there are positive constants $c_{1}$ and $c_{2}$ such that
$$\forall t\geq 0 : \,\,\,\,\,\,\ \mathbb P\{|X-\mathbb EX|\geq t\}\leq c_{1}e^{-c_{2}t^{2}}...
- 21
2
votes
1
answer
436
views
Best approximation of a compactly supported density by a single Gaussian
Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow.
Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability ...
- 710
3
votes
2
answers
2k
views
The norm of isotropic sub-Gaussian random vector may not be sub-Gaussian
Suppose $X$ is a isotropic sub-Gaussian $n$-dimensional random vector (i.e. $EXX^T=I_n$, and for any unit vector $u$,$\|\left<X,u\right>\|_{\psi_2}\le K$). It is said that $\|X\|_2-\sqrt n$ may ...
- 601
3
votes
2
answers
818
views
Is there a name for "splitting a probability distribution into independent components"?
Suppose I have a random variable $\theta=(\theta_1,\dotsc,\theta_n)$; where the $\theta_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi_1,\dotsc,\phi_k)$, where $\hat\...
- 267
2
votes
0
answers
100
views
Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes
Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
- 560
3
votes
1
answer
498
views
Strictly Proper Scoring Rules and f-Divergences
Let $S$ be a scoring rule for probability functions. Define
$EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$.
Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a ...
- 631
3
votes
3
answers
477
views
A clean upper bound for the expectation of a function of a binomial random variable
I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.
- 101
0
votes
1
answer
101
views
How to find a special random variable? [closed]
Suppose random variables $X_1$ and $X_2$ have the same distribution under P, $Y_1$ is an arbitrary random variable,let $Z_1:=X_1+Y_1$.Can we find a r.v. $Y_2$ which has same distribution as $Y_1$,such ...
- 13
3
votes
1
answer
694
views
Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture
Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...
- 167
1
vote
0
answers
56
views
About a class of expectations
Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
- 2,246
2
votes
1
answer
847
views
Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix
Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (not necessarily idempotent)
what is the distribution of ${y^T M y}$?
is there a high probability bound on $|{y^T M y}|$?
Most ...
- 123
4
votes
1
answer
699
views
Rate of decay in the multivariate Central Limit Theorem
The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S_n=X_1+\cdots+X_n$ of independent random variables $...
- 2,288
5
votes
1
answer
997
views
Variance of sum of $m$ dependent random variables
I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here.
Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...
- 339
1
vote
0
answers
221
views
Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?
Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls.
The red balls and the white balls are randomly distributed across the bins (that is, for ...
- 53
2
votes
1
answer
106
views
How to efficiently sample uniformly from the set of $p$-equipartitions of an $n$-set?
I have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all equipartitions of $n$ in $p$ sets; i.e., in sets of equal size $\...
- 1,372
3
votes
2
answers
309
views
Couplings on empirical distributions
For a problem I've been working on, I'm thinking about couplings between true and empirical distrubutions.
I have two datasets $S$ and $T$ with underlying measures $\mu_S,\,\mu_T$. And then I have ...
- 383
1
vote
1
answer
239
views
Probability of two Points being divided by an high-Dimensional Hyperplane
I have two points $x_1,x_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the ...
- 899
0
votes
0
answers
268
views
Taylor series expansion of quantile function
Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...
- 250
1
vote
1
answer
1k
views
Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers
We consider the two distributions
$$
p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I),
$$
where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...
- 1,127
23
votes
5
answers
2k
views
Maximizing the expectation of a polynomial function of iid random variables
Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$.
Question 1. What is ...
- 980
1
vote
1
answer
137
views
Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$
Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the ...
- 6,853
1
vote
1
answer
90
views
Expectation of random variables coincides
Let $Y_1:=(X_i)_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent.
We can then also define the shifted sequence $Y_2:=(X_{i+1})_{i \...
- 13
3
votes
4
answers
451
views
Solution of a 2D Recurrence sequence
Can we solve the following recurrence relation:
$$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$
with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$
I encountered this ...
- 1,495
1
vote
1
answer
88
views
Independence of r.v.'s following a distribution that is the ratio between complex Gaussian and Chi-square r.v.'s
Given the following two R.V.s
$$z_1 = \frac{x_1}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$
and
$$z_2 = \frac{x_2}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$
where $x_i \sim \mathcal{CN}(0,a), \forall i$...
1
vote
1
answer
89
views
Correlation between r.v.'s following a distribution that is the ration between complex Gaussian and Chi-square r.v.'s
Given the following two R.V.s
$$z_{1} = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
and
$$z_{2} = \frac{x_{2}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
where $x_{i} \sim \mathcal{CN}(...
1
vote
0
answers
52
views
Stochastic Control: Markovian restriction
Consider a stochastic control problem,
$$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big] $$
where $X_t$ is a weak solution to the SDE
$$dX_t = C_t dB_t, \quad X_0 = x \...
- 553
1
vote
1
answer
186
views
Expected norm of linear maps
I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from ...
- 899