Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

Filter by
Sorted by
Tagged with
-4 votes
0 answers
65 views

Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent [closed]

Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$. Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=...
user avatar
-1 votes
2 answers
93 views

Cumulants of a sequence of variables with zero mean and variance

Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
user avatar
  • 11
0 votes
1 answer
62 views

Functional relationship between two quantities

Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by $$ \alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...
user avatar
  • 5,580
6 votes
1 answer
409 views

Max decoupling inequality

Let $X_1,\ldots,X_n$ be $\{0,1\}$-valued random variables drawn from some joint distribution. Let $\tilde X_1,\ldots,\tilde X_n$ be their independent version: $\mathbb{E}X_i=\mathbb{E}\tilde X_i$ for ...
user avatar
3 votes
2 answers
125 views

Continuity of Radon transform w.r.t the angle

Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by $$ R[f](w,b) := ...
user avatar
  • 5,580
0 votes
0 answers
72 views

$L^p$ inequality for "positively correlated" random variables

Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
user avatar
3 votes
0 answers
85 views

Does smoothing a non-log-concave distribution make it more log-concave?

Suppose that $p$ is a density on $\mathbb{R}^d$ that is $C^2$ and nonzero everywhere, and such that the Hessian of its negative logarithm is lower bounded: $$-\nabla^2 \ln p\succeq L$$ for some matrix ...
user avatar
3 votes
0 answers
48 views

Random assignment problem under multinomial or Poisson distribution

We place $m$ balls at random (uniformly) inside $n^2$ urns arranged as a $n \times n$ square. Then we must choose $n$ urns, such that no two urns belong to the same row or column, with the objective ...
user avatar
  • 298
1 vote
1 answer
39 views

Estimation of Lévy measure of ID distribution

Suppose that the positive random variable $X$ is infinitely divisible and supported on $\mathbb R_+$. Due to Lévy-Khintchine, its moment generating function then writes : $$M(t) = \mathbb E\left(e^{tX}...
user avatar
  • 621
0 votes
0 answers
80 views

Stopping times for martingale

The nonnegative integer set is denoted by $\mathbb{Z}_+$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$ be an increasing sequence ...
user avatar
  • 51
1 vote
0 answers
91 views

On probability of coprimality of a list of numbers

We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$. Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
user avatar
  • 12.8k
0 votes
0 answers
43 views

On the Markov property of a limit process

Let $E$ be a locally compact separable metric with countable base. We consider a sequence of Hunt processes $\{X^{(n)}\}_{n \in \mathbb{N}}$ on $E$. That is, each $X^{(n)}=(\{X_t^{(n)}\}_{t \in [0,\...
user avatar
  • 703
2 votes
1 answer
118 views

Verify if array is orthogonal

This is a repost from the computer science stackexchange. The question has been offered a bounty, but received no answers. Therefore, I would like to ask this question here. Orthogonal arrays often ...
user avatar
  • 470
0 votes
0 answers
22 views

L^2 approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
user avatar
2 votes
0 answers
37 views

techniques in studying moments of shifted integral process $\mu(T_{a},T_{a}+t)$

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...
user avatar
  • 1,378
7 votes
2 answers
185 views

On permanent of a square of a doubly stochastic matrix

Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...
user avatar
1 vote
0 answers
41 views

Counting overlaps of random intervals

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...
user avatar
  • 1,378
1 vote
0 answers
122 views

A basic formula for the falling factorial

Whis is a question I asked on Math.SE, but didn't get any response. Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e. $$ X(\omega): = \sum_{A \in \...
user avatar
  • 1,453
0 votes
0 answers
29 views

How to use Itô's formula to show that $ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\left<\mathbf{J}x_u, x_t\right>]du+\frac{1}{N}\sum x_t^i(B_s^i-B_t^i) $?

I am reading a lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it shows that for $s\ge t$, $$ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\langle\mathbf{J}...
user avatar
  • 156
3 votes
2 answers
118 views

On finding an upper bound on the error of a sparse approximation

I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer. Original question: https://math.stackexchange.com/...
user avatar
0 votes
0 answers
18 views

Vector Norms of Sub-Gaussian Matrix Multiplication?

Let $X, Y$ be $n\times n$ matricies with i.i.d. sub-Gaussian entries. I am interested in tail bounds for $\lVert XY\rVert$, where $\lVert\cdot\rVert$ is a norm on $\mathbb{R}^{n^2}$, i.e. I want tail ...
user avatar
  • 702
4 votes
1 answer
134 views

Population growth with good and evil children - probability good outnumbers evil

Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those ...
user avatar
1 vote
1 answer
76 views

Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
user avatar
  • 5,580
1 vote
0 answers
46 views

Sobolev variant of Wasserstein space

Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
user avatar
4 votes
1 answer
210 views

How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?

I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging. On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...
user avatar
  • 156
0 votes
1 answer
128 views

Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions

We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
user avatar
1 vote
0 answers
30 views

Handling sums of correlated random variables with a directed path structure

Recently, I've been seeing random variables with the following correlation structure based on directed paths on a graph. For example, there are $16$ directed paths "directed downwards from the ...
user avatar
1 vote
1 answer
54 views

Non-independent Sub-gaussian variables and concentration

Let $g \in R^{d}$ have $iid$ Gaussian components. Let $a \in R^{d}$, and let $b \in R^{d}$. be arbitrary vectors. Consider the random variable $Y_{g,g}:= \frac{1}{n}\langle g,a \rangle \langle g, b \...
user avatar
  • 117
0 votes
1 answer
48 views

Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?

Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?
user avatar
  • 11
0 votes
0 answers
61 views

Converse to Cameron-Martin theorem

It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....
user avatar
5 votes
1 answer
107 views

Second Skorokhod embedding in high dimensions

The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is ...
user avatar
  • 651
0 votes
0 answers
29 views

Minimum eigenvalue of covariance matrix under probability constraint

Consider a random (column) vector $X\in \mathbb{R}^d$. I am interested in the quantity $$ \Lambda(\alpha)=\inf_{E\in \mathcal{B}(\mathbb{R}^d), \ P(X\in E)\ge \alpha} \lambda_{\min}\left(E[XX'1\{X\in ...
user avatar
  • 377
2 votes
1 answer
68 views

Chung's law of the iterated logarithm for Brownian motion

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...
user avatar
  • 121
0 votes
0 answers
45 views

A sufficient condition for the decomposition of a bounded random vector

Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{(x_1,x_2,\ldots,x_n)|\sum_{i=1}^m{\bf{a}}_ix_i,x_i \in [-1,1]\}$, where ${...
user avatar
1 vote
0 answers
26 views

A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?

Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI) $$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$ with LSI constant $\...
user avatar
0 votes
0 answers
60 views

Expectation of edge weights on the complete graph, Part 2

This question concerns the same basic set-up as my previous question: Expectation of edge weights on the complete graph In that question an answer was given which shows that the expected value is as ...
user avatar
2 votes
0 answers
50 views

Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that $$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$ I'm looking for a similar (asymptotic) bound for asymptotically normal ...
user avatar
  • 213
0 votes
0 answers
39 views

Total variation convergence of eigenvalues in the bulk of the GUE?

Let $\lambda_{k(n)}$, $k(n) / n \to a \in (0,1)$, be an eigenvalue in the bulk of the $n \times n$ Gaussian Unitary Ensemble (GUE) normalized so that its spectral measure converges to the semi-circle ...
user avatar
-1 votes
0 answers
178 views

Open problems in derivatives, options and finance [closed]

I would like to ask what kind of problems remain open in finance. In particular, those concerning pricing derivatives, computing the risk, etc. It could be really of great help if you can recommend to ...
user avatar
1 vote
1 answer
131 views

Expectation of edge weights on the complete graph

Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...
user avatar
2 votes
1 answer
87 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
user avatar
  • 162
4 votes
1 answer
70 views

Distance between trunctated random walk and its normal form

I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$ $$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...
user avatar
  • 43
3 votes
3 answers
465 views

How close are two Gaussian random variables?

Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
user avatar
1 vote
0 answers
68 views

On the closedness of a certain subset of $\mathbb R$

Let $\mu$ be a probability measure on measurable space $X=\mathbb R^n$ (euclidean), and let $F$ be a family of $\mu$-measurable functions $X \mapsto \mathbb R$ which are uniformly bounded, i.e $b:=\...
user avatar
  • 5,580
0 votes
0 answers
61 views

Extending proofs from Lebesgue measure to non-atomic (probability) measures [closed]

On $\mathbb R^n$, is there a relationship between non-atomic probability measures and the Lebesgue measures ? What kinds of results about non-atomic measures have the same proof as for Lebesgue ...
user avatar
  • 5,580
2 votes
1 answer
62 views

Compactness of the integral of a set-valued function

Let $X$ be a compact space (e.g. a compact subset of $\mathbb R^n$) and $P$ be a probability measure on $X$. Let $A$ be a compact subset of some $\mathbb R^d$. Finally, let $F$ be the collection of $P$...
user avatar
  • 5,580
1 vote
1 answer
54 views

Independent Sums and Orlicz Norms

Let $X_{i}$ be a collection of iid random variables of cardinality $n$, and let $S_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i}.$ Let $|| X||:=\inf_{B}\{E[\exp(X/B)-1]\leq 1\}$. This is the so-called sub-...
user avatar
  • 117
3 votes
0 answers
33 views

Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient

Consider the SDE below $$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$ where $X_0>0$ is square integrable, $b:\mathbb R_+\...
user avatar
  • 804
0 votes
0 answers
29 views

How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example. I have a weighted probability list A : 0.15 B : 0.15 C : 0.15 D : 0.1 E : 0.1 F : 0.1 G : 0.1 H : 0.075 I : 0.075 ...
user avatar
7 votes
2 answers
182 views

Does entropy of the random walk control the return probability

Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n_x(y)$ is the probability ...
user avatar
  • 4,214

1
2 3 4 5
152