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.
8,619
questions
7
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0
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110
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Cohomology dimensions are well approximated by Gaussian for multiply-fibered manifolds ? (Topological central limit theorem)
Consider some manifold $M$ say compact smooth. Let $b_i$ be its Betti numbers (non-zero), i.e. its cohomology dimensions.
Assume $M$ can be subsequently fibered by many manifolds, i.e. there is $ M_{...
1
vote
0
answers
31
views
Concentration result for self-normalized empirical process
In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
1
vote
0
answers
30
views
Example of $F\in W_0^{1,2}$ a.s. so that the law of $F+B$ is equivalent to that of $B$ but DD exponential isn't integrable?
Is there an explicit example of progressively measurable $F=\int_0^\cdot f(s) ds\in W_0^{1,2}(0,1)$ a.s. so that the law of $F+B$ on $(0,1)$ is equivalent to that of a Brownian motion $B$ on $(0,1)$ ...
2
votes
0
answers
35
views
Including fixed-time transitions into a continuous time Markov chain system
I have system which is mostly described by a CTMC (Continuous-time Markov chain) with a single absorbing state and a large but tractable and sparse transition matrix. However, at a fixed set of "...
-3
votes
0
answers
41
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?
-1
votes
0
answers
26
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:...
3
votes
0
answers
109
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 $c(n,d)>0$ ...
10
votes
1
answer
352
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 ...
0
votes
0
answers
37
views
Hopfield neural network [closed]
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 ...
3
votes
1
answer
77
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 $\...
1
vote
1
answer
137
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\...
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 $\...
0
votes
0
answers
27
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 ...
2
votes
1
answer
58
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(\...
2
votes
0
answers
36
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 ...
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 $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
2
votes
1
answer
71
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 \...
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 ...
0
votes
1
answer
65
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 ...
1
vote
0
answers
98
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 ...
0
votes
0
answers
63
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 ...
1
vote
1
answer
85
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(\...
-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 ...
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 ...
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 ...
-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(\...
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 ...
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
\...
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+...
0
votes
1
answer
145
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 ...
-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 ...
0
votes
0
answers
65
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 ...
6
votes
0
answers
69
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 $n$-...
1
vote
2
answers
225
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
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...
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\...
2
votes
1
answer
225
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 ...
7
votes
0
answers
167
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 ...
4
votes
2
answers
262
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(\...
0
votes
1
answer
89
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 ...
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\...
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 ...
-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, ...
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. ...
2
votes
0
answers
66
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
$$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
2
votes
1
answer
238
views
The probability that iid draws from a mean zero random variable sum to zero
Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are ...
-2
votes
0
answers
30
views
Do we need to assume that $y$ is bounded or subgaussian?
Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$
For a real-valued function $f$ on $\...
8
votes
1
answer
332
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 $p$...
1
vote
1
answer
180
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 $1/2$....
0
votes
0
answers
64
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 ...
1
vote
1
answer
115
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 \...