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
0 votes
0 answers
7 views

Including fixed-time transitions into a continuous time Markov chain system

I have system which is mostly described by a CTMC with a single absorbing state and a large but tractable and sparse transition matrix. However, at a fixed set of "switch times" {t1,t2,...,...
Bianca's user avatar
  • 1
-1 votes
0 answers
11 views

the distribution of a stopping time of a Brownian motion

Is there example of a stopping time of a standard brownian motion which has discontinuous distribution? is there any general result for such stopping time?
user524762's user avatar
-1 votes
0 answers
21 views

Smooth sensitivity for high-dimensional problems

Smooth sensitivity comes from the following paper: https://dl.acm.org/doi/pdf/10.1145/1250790.1250803?casa_token=YKSULDNu3Z8AAAAA:...
jkfds's user avatar
  • 9
2 votes
0 answers
87 views

Anti-concentration of polynomials on Haar measure

Let $X\in\mathbb{R}^n$ follow the Haar measure (i.e. uniformly distributed over the unit sphere), and $P$ be a degree-$d$ polynomial such that $\mathrm{Var}[P(X)]=1$. Are there constants $c(n,d)>0$ ...
Wei Zhan's user avatar
  • 193
8 votes
1 answer
231 views

The drunken blind man’s walk

Consider a drunk, blind man starting in the middle of the two dimensional open unit ball. At each turn, the man chooses a direction to move a step of size $\delta > 0$ in. Unfortunately, he is very ...
Nate River's user avatar
  • 4,762
0 votes
0 answers
36 views

Hopfield neural network

I started research on Hopfield neural network, but I faced some problems, my questions are 1 Why do we use continuous fractional calculus in neural Hopfield and what are the advantages? 2 Why do we ...
Bady's user avatar
  • 1
3 votes
1 answer
69 views

A concentration inequality derived from Freedman’s inequality

Freedman’s inequality is a well-known concentration inequality of martingale difference sequence: Let $(Z_t)_{t \leq T}$ be a real-valued martingale difference sequence adapted to filtration $\...
Mixi Andrew's user avatar
1 vote
1 answer
119 views

Chebyshev's inequality for Poisson distribution

Reading an old Richard Karp paper, in which he mentions this argument "Application of Chebyshev's inequality yields the result that, if $X$ is Poisson distributed with mean $\lambda$, then $E(X\...
OmarR's user avatar
  • 67
0 votes
0 answers
17 views

reference request: product measures defined by a subsequence of measures

Suppose $\{\mu_n\}_{n\in\mathbb{N}}$ is a sequence of pairwise equivalent probability measures, each of which is defined on $\mathbb{R}$. Let $\bigotimes_n\mu_n$ be the product measure defined on $\...
Sanae Kochiya's user avatar
0 votes
0 answers
24 views

Limiting value of trace of resolvent matrix involving two independent Wishart random matrices

Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that $$ d/n_k \to \phi_k \in (0,\infty). $$ Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
dohmatob's user avatar
  • 6,686
2 votes
1 answer
56 views

From convergence of sequences to uniform convergence in probability

For $n=1, 2,\ldots$ consider a sequence of sets of ascending integers $I_n=\{\underline{i}_n,\underline{i}_n+1, \ldots, \overline{i}_n\}$, with $\underline{i}_n \to \infty$ and $\underline{i}_n=o(\...
Jack London's user avatar
2 votes
0 answers
35 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 ...
Tom's user avatar
  • 716
1 vote
0 answers
27 views

Deterministic multifractal measure with quadratic singular spectrum?

For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$ ...
MikeG's user avatar
  • 665
2 votes
1 answer
69 views

Gradient flows and particle representations

I was looking into gradient flows and their particle representations, mostly in the context of probability. A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
CComp's user avatar
  • 111
2 votes
0 answers
46 views

Does this filtration have a name?

In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
Mushu Nrek's user avatar
0 votes
1 answer
62 views

Stationary distribution of AR(1) processes and Lyapunov central limit theorem

Let $X_t$ follow the following AR(1) process: $$ X_t=\rho X_{t-1}+e_t $$ in which $|\rho|<1$ and $e_t$ is iid noise term with density $f$, mean $0$ and finite moments up to a certain order. I am ...
Lemma1's user avatar
  • 155
1 vote
0 answers
96 views

Density of absolutely continuous measures on a Polish Space

Consider the set of all probability measures on a Polish space $X$ (equipped with the Borel $\sigma$-field $\mathcal{B}(X)$). I am wondering if there exist conditions under which a subset of measures ...
d.k.o.'s user avatar
  • 185
0 votes
0 answers
62 views

Inequality related with log-concave distributions

Fix any $n$-dimensional unit vector $\mathbf v$. Let $\mathbf x$ be a random vector following the $n$-dimensional standard normal distribution. It has been shown (Analysis of Perceptron-Based Active ...
entechnic's user avatar
  • 141
1 vote
1 answer
84 views

Concentration inequalities for heavy-tailed distributions

Suppose $X_1,...,X_N$ are $N$ i.i.d random variables with heavy tailed distributions. For example, $E[X_i^p]\leq 1$ for some $p\geq 1$. Are there some concentration inequalities to bound the tail $$P(\...
jkfds's user avatar
  • 9
-1 votes
0 answers
34 views

Expected value of a Stochastic process

Consider a discrete stochastic process $\{X_t\}_{t \in T}$ with the following properties. Each $t \in T$ has a value $v(t) \in \mathbb{R}_{+}$ and the value is added to the overall value conditioned ...
John's user avatar
  • 161
1 vote
0 answers
130 views

Ask assistance for finding K. Sato - Lévy Processes on the Euclidean Spaces

The paper me and my professor want is called K. Sato (1995) Lévy Processes on the Euclidean Spaces, Lecture Notes, Institute of Mathematics, University of Zurich. I tried to find the paper on the ...
Zoël Li's user avatar
3 votes
0 answers
64 views

Is there a way in which "space" of random variables on $\mathbb{R}$ is canonically a coaugmented coalgebra?

Consider the "space" of random variables with finite expectation on $\mathbb{R}$ in the following sense: we fix the Borel $\sigma$-algebra on $\mathbb{R}$, and put random variables in ...
Daigaku no Baku's user avatar
-1 votes
0 answers
45 views

Accuracy of the definition of the space-time white noise

We have seen that there exists $\overline{\xi}:\Omega \to \mathcal{E}'$ ($\mathcal{E}'$ is the space of tempered distributions) which usually replaces the space-time white noise (indexed by $L^2(\...
mathex's user avatar
  • 387
0 votes
0 answers
80 views

Martingale defined by an integral

Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
mathex's user avatar
  • 387
1 vote
1 answer
108 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 \...
Ben's user avatar
  • 21
1 vote
0 answers
57 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+...
Ben's user avatar
  • 21
0 votes
0 answers
18 views

Distribution function for multiple offsprings [closed]

I am taking a course on branching processes and one of the questions was the following: \begin{align*} \text{Given:} \quad & X_0 = 10, \quad Y_0 = 1 \\ \text{Distribution of offsprings:} & \\ \...
Steve k97's user avatar
0 votes
1 answer
141 views

Construction of random tempered distributions

Let $(\xi_\phi)_{\phi \in L^2(\mathbb{R}_+ \times \mathbb{R}^d,\lambda_d)}$ be a collection of centered Gaussian processes on a probability space $(\Omega,\mathcal{F},P)$ such that $$\forall \phi \in ...
mathex's user avatar
  • 387
-1 votes
0 answers
80 views

Stochastic dominance for (~)random harmonic series

$\DeclareMathOperator\Pr{Pr}$Consider the series $\sum_n^\prime a_nR_n$, where $a_n=\frac{(-1)^n}{n+c}$ for some constant $c\in(0,1)$ and $\{R_n\}$ denotes a sequence of i.i.d. Bernoulli random ...
Ben's user avatar
  • 21
0 votes
0 answers
63 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 ...
BayesRayes's user avatar
6 votes
0 answers
68 views

Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)

Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
ABIM's user avatar
  • 4,979
1 vote
2 answers
220 views

Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
0 votes
1 answer
55 views

Does point process ordering ever imply conditional intensity ordering?

Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
jdods's user avatar
  • 213
2 votes
1 answer
223 views

If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?

Although very useful in the Gaussian (or other infinite moment) setting, Kolmogorov continuity criterion is non optimal in the finite moment setting. For example, let $X(t)=Zt$ where $Z$ is a random ...
user479223's user avatar
  • 1,221
6 votes
0 answers
155 views

Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
Alexander Chervov's user avatar
4 votes
2 answers
259 views

Lower bounding a partition-related sum

We say the $\mathbb{N}$-valued, non-increasing, eventually zero sequence $\lambda=(\lambda_1\geq\lambda_2\geq\cdots)$ is a partition of $N$ if $|\lambda|:=\sum_{k\geq 1}\lambda_k=N$, and denote $m_k(\...
MikeG's user avatar
  • 665
0 votes
1 answer
88 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 ...
Sanket Agrawal's user avatar
0 votes
0 answers
70 views

Some new questions on Rademacher complexity

For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable. Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
Hao Yu's user avatar
  • 185
1 vote
0 answers
109 views

Random partition of an interval – Dirichlet distributed?

Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$. What is the distribution ...
gusl's user avatar
  • 47
-1 votes
0 answers
39 views

How to modeling continuous batching in large-scale inference with queuing theory approach?

I want to model continuous batching in large model inference problems, but my knowledge in data theory is insufficient, and I haven't found an appropriate queuing theory model to use for modeling. So, ...
oleotiger's user avatar
3 votes
0 answers
77 views

Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...
Luís Ferreira's user avatar
2 votes
0 answers
61 views

Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
Vladimir Zolotov's user avatar
2 votes
1 answer
235 views

The probability that iid draws from a mean zero random variable sum to zero

Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are ...
James Propp's user avatar
  • 19.4k
-2 votes
0 answers
30 views

Do we need to assume that $y$ is bounded or subgaussian?

Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$ For a real-valued function $f$ on $\...
Hermi's user avatar
  • 274
8 votes
1 answer
330 views

Optimally betting a beta-biased coin

This question is inspired by How to optimally bet on a biased coin? by Nate River but generalized slightly. I decided the generalization might be interesting enough to be its own question. A number $p$...
Will Sawin's user avatar
  • 134k
1 vote
1 answer
178 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 $1/2$....
odile's user avatar
  • 65
0 votes
0 answers
63 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 ...
Dimitri's user avatar
1 vote
1 answer
113 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 \...
odile's user avatar
  • 65
1 vote
1 answer
101 views

Does a random matrix over $\mathbb{Z}_q$ map linearly independent vectors to statistically independent vectors?

Suppose $A \in \mathbb{Z}_q^{n \times m}$ is a random $n \times m$ matrix whose entries are i.i.d. uniform over $\mathbb{Z}_q=\mathbb{Z}/q\mathbb{Z}$, where $q\geq2$. Let $\mathbf{x}_1, \ldots, \...
aleph's user avatar
  • 503
1 vote
1 answer
82 views

A Kolmogorov inequality for sums of contiguous subsequences

If $X_1, \ldots X_n$ are independent real-valued random variables such that $E[X_k] = 0$ and $E[X_k^2]$ is finite for each $k$, Kolmorogov's inequality gives an upper bound on $P[\max_{1\le k \le n}|...
Rob Arthan's user avatar

1
2 3 4 5
173