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.

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Reference request - logarithmic average

Consider a set $A\subseteq\mathbb{N}$. Consider an arithmetic function $a(n):\mathbb{N}\to\mathbb{C}$. I am looking for notation which describes the following:\begin{equation}\frac{\sum_{n\in A}\frac{...
alidixon222's user avatar
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Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables

Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
Marco Max Fiandri's user avatar
4 votes
1 answer
209 views

Randomly removing length 1 intervals in an interval (a fragmentation process)

Short version: Start with a closed interval of length $t>0$ and repeatedly remove a random and uniformly distributed subinterval of length $1$ so long as this is possible. For $t$ large, what is ...
Gro-Tsen's user avatar
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3 votes
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Concentration of measure on spheres with respect to a unitary of trace approximately zero

Cross-posted from MSE, where it hasn’t received any answer yet: This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
David Gao's user avatar
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Probability of sequence of estimated values belonging to a set

Suppose I have estimated residuals of an ARCH(1) process: $\hat{\varepsilon}_1, \dots, \hat{\varepsilon}_n$ from the sample of length $n$. On the other hand, I have "true" residuals: $\...
Grigori's user avatar
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Characterization of Fellerian kernels

This question concerns Feller Markov kernels, similar to Vanessa's question. Terminology By 'Markov kernel' $N:E\to F$, we adopt exactly the same definition as Vanessa, with the exception that $E,F$ ...
Ano2Math5's user avatar
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estimating probabilities like mortality rates among specific demographics is impossible? [closed]

Russell had a similar view to this that i will summarize it in the following example: What is the probability that an English man who reaches the age of sixty will die within one year, and the first ...
Hadibinalshiab's user avatar
1 vote
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In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?

Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements: $\lambda$ being a random large prime such as $w^\lambda > 2\times m$ $1 < n < m−1$. m is ...
user2284570's user avatar
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Formalization of sample convergence

Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that $$X_i = \sigma_i z_i,$$ where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\...
Grigori's user avatar
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Limiting distribution of separated points in a unit square

Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start: For $i\in\{1,\dots,n\}$: Sample a random point $X$ in the unit square. If $X$ is a distance at least $r$ ...
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White noise, stochastic convolution: $0$–$1$ law of a stopping time

Let $\mathscr{C}^\alpha:=B_{\infty,\infty}^{\alpha}$ be the Besov space with the usual norm and let $C_T\mathscr{C}^\alpha:=C([0,T],\mathscr{C}^\alpha)$ the space of continuous functions from $[0,T]$ ...
mathex's user avatar
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Directions of differentiability of log-concave measures with infinite-dimensional support

I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
iolo's user avatar
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Convergence of probabilities imply convergence of joint probability

Context: Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\...
Grigori's user avatar
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1 answer
68 views

Difference of probabilities of two random vectors lying in the same set

Suppose I have to random vectors: $$\mathbf{z} = (z_1, \dots, z_n)^T, \quad \mathbf{v} = (v_1, \dots, v_n)^T$$ and set $A \subset \mathbb{R}^n$. I want to find an upper bound $B$ for the following ...
Grigori's user avatar
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A question about one Malliavin derivative calculation

Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
tsnao's user avatar
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Expectation of $B_u \operatorname{argmax}_t B_t$

This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here. Yesterday I asked a question about the joint law of ...
tsnao's user avatar
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8 votes
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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_{...
Alexander Chervov's user avatar
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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....
WeakLearner's user avatar
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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 $(0,1)$ ...
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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 "...
Bianca's user avatar
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the distribution of a stopping time of a Brownian motion [closed]

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
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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
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3 votes
1 answer
171 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
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10 votes
1 answer
439 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
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3 votes
1 answer
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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
154 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
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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
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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 ...
dohmatob's user avatar
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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(\...
Jack London's user avatar
2 votes
0 answers
39 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
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1 vote
0 answers
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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
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2 votes
1 answer
75 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
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2 votes
0 answers
47 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
69 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
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1 vote
0 answers
105 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
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0 answers
68 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
93 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
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-1 votes
0 answers
40 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
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1 vote
0 answers
131 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
68 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
0 votes
0 answers
84 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
  • 391
1 vote
1 answer
109 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
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0 answers
60 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
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0 votes
1 answer
149 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
  • 391
-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
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0 votes
0 answers
66 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
70 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
  • 5,019
1 vote
2 answers
233 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
231 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
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