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.
9,021 questions
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What does a product of many Gaussian matrices converge to?
Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$.
Is ...
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Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$
Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$.
Define
$$
F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big]
$$
If $\|x\|_\infty \...
- 420
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Size doubling amoeba
Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.
A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p &...
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answers
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Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures
How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
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0
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73
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Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
2
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1
answer
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Proof of the Dunford-Pettis theorem in the context of probability spaces
I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
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A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 51
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Square-integral involving Brownian bridge
Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of
$$
\int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt?
$$
As noted I am interested ...
- 2,207
-1
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Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\mathbf{x})$
Let $f(\mathbf{x})=f(x_1,x_2,\dotsc,x_N)$ with $N>2$ be a real and continuous function and $f(\mathbf{x})\ge f_0$ for any $\mathbf{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\dotsc,x_N$ be the i.i.d. ...
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+100
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 420
5
votes
1
answer
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views
Moment Bounds on Hölder norms of stochastic processes
It is relatively easy to show that a stochastic process is Hölder continuous using Kolmogorov continuity theorem link text. But how does one obtain a bound $\mathbb{E} \left\Vert u\right\Vert _{\gamma}...
CommunityBot
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1
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On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
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1
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Given iid $w_1,\dotsc,w_N \sim N(0,1/d)$ iid, find a simple matrix $A$ s.t $\|aa^T-A\|_\text{op}\to0$, where $a_i := E_{G \sim N(0,1)}[f(\|w_i\| G)]$
Let $d$ and $N$ be two large comparable integers, for example assume
$$
N,d \to \infty, \quad d/N \to \gamma \in (0,\infty).
$$
Let $w_1,\dotsc,w_N$ be iid from $N(0,(1/d)I_d)$ and let $f:\mathbb R \...
- 12.9k
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answers
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views
Can conditional distributions with respect to a sufficient sub-$\sigma$-algebra be represented by a single Markov kernel?
Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be ...
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votes
1
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Surjectivity of pushforward on image
Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
34
votes
3
answers
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Intrinsic significance of differential entropy
Many commentators (e.g. Jaynes, Rota) argue that the notion of "differential entropy" is problematic (as commonly defined by $ h(X) = \int ( \log\frac{1}{p(x)} ) p(x) \, dx $, where $X$ is a random ...
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1
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Proving bound on expectation of likelihood ratio involving mixtures
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 128k
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1
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Convergence of edge eigenvalues for Gaussian matrices
I am reading this lecture note.
I have a difficulty in understanding the third section in chapter 6. Particularly, in Theorem 4.1, they claimed that
Let $X$ be a Gaussian Wigner matrix satisfying ...
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3
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How can I randomly draw an ensemble of unit vectors that sum to zero?
Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question:
Is there a way to sample ...
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0
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70
views
Exponential decay for Bernoulli percolation with a ghost field
Background on Bernoulli percolation:
Consider the hypercubic lattice $\mathbb{Z}^d$ as a graph with vertices as point of integer coordinates and edges between points of distance 1.
Now, delete every ...
- 1,054
14
votes
1
answer
1k
views
Berry Esseen type result for probability density functions
Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_1) = 0, E(X_1^2) = \sigma^2 >0, E(|X_1|^3) = \rho < \infty$.
Let $Y_n = \frac{1}{n} \sum_{i=1}^n X_i$ and let us note $F_n$ (resp. $\...
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vote
0
answers
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Quantitative multivariate CLT from quantitative CLT of linear combinations
Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
- 111
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0
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Characterisation of a family of continuous martingales
I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that
$$X_0=0\quad \mbox{ and } \quad\...
- 63.9k
2
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2
answers
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views
How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
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1
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views
Median of cardinality of set union
Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
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1
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What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
- 1,904
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votes
0
answers
38
views
Bounding the error of a truncated moment problem
Let $\{x_{i}\}_{i=1}^{\infty}$ be a non-increasing sequence of non-negative real numbers, and let $\{y_{j}\}_{j=1}^{B}$ be a non-increasing sequence of non-negative real numbers, where $B$ is a finite ...
- 433
5
votes
1
answer
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Asymptotic distribution of the extreme, standardized order statistics of uniform distribution?
Let $\{U_{k, n}\}_{k=1}^n$, denote the order statistics of a sample of $n$ iid uniform $[0, 1]$ variates.
Note that, marginally $U_{k, n}$ is distributed $\mathrm{Beta}(k, n+1 -k)$.
Therefore, let us ...
CommunityBot
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1
answer
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Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1
Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...
CommunityBot
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0
answers
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Different definition of Feller semi-group
(This is a crosspost of a question on MathStackExchange which did not receive any answer.)
Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ ...
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9
votes
2
answers
429
views
Hermite–Fourier expansion for the median
Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier ...
- 128k
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votes
1
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673
views
A random variation on Pólya's orchard problem
Pólya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e....
- 105k
3
votes
2
answers
545
views
Intensity and compensator for a jump process
Set-up and assumptions. Let $(\mathscr{F}_t, t \geq 0)$ be a right-continuous complete filtration. Let $(X_t, t\geq 0 )$ be a pure jump $\mathbb{R}$-valued process with unit jumps, that is,
$$
X_t = \...
2
votes
1
answer
328
views
Matrix Bernstein's inequality: from tail probability to expectation
Let $X_i$ be independent, mean zero, $n\times n$, symmetric random matrices. $\|X_i\|\leq K$ almost sure for $\forall I$.
We have matrix Bernstein's inequality for the tail probability as follows
$$\...
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Variance of bins for N balls into M bins [closed]
If I throw N balls independently into M bins with uniform probability, the expected mean of the M bins is N/M balls.
What is the expected variance of the M bins?
I was thinking of what bin size I ...
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2
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Simple proof of sharp constant in DKW inequality
The DKW inequality says that if $F_n$ is the empirical CDF corresponding to real-valued random variables $X_1, \dots, X_n$ distributed identically and independently from a distribution with CDF $F$, ...
CommunityBot
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2
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1
answer
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Lower bound in the singularity of random Bernoulli matrices
Let $A_n$ be a random $n \times n$ matrix with entries in $\{-1, +1\}$. As usual, "random" here means with respect to the uniform measure over such matrices.
The strong version of the ...
- 188k
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votes
1
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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\...
CommunityBot
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votes
3
answers
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Is there a probability theory developed in intuitionistic logic?
Since Boole it is known that probability theory is closely related to logic.
According to the axioms of Kolmogorov, probability theory is formulated with a (normalized)
probability measure $\mbox{...
CommunityBot
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Asymptotics of number of running maxima of iid random variables
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.
Given a realisation $\omega$ of the random variables, we say that $X_i (\...
- 128k
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Convergence of random functions
Suppose I have a sequence of random continuous functions, $f^{n} : [0, t] \to \mathbb{R}$. Suppose there also exists a random continuous function, $f: [0, t] \to \mathbb{R}$, defined on the same ...
- 10.3k
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1
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379
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Uniqueness of maximizer of dual Kantorovich problem with quadratic(or any strictly convex) cost
I am considering the optimal transport problem under the setting $X=\mathbb{R}^n$, $\mu,\nu\in\mathcal{P}(X)$ be two probability measures, and the cost function is $c(x,y)=|x-y|^2$. We know from ...
CommunityBot
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votes
1
answer
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Components of a Gram matrix and its eigenvalues
The Gram Matrix is defined as $$\sum_{i=1}^n X_iX_i^T,$$ where $X_i$ is drawn from the unit sphere based according to some continuous distribution (Relation between eigenvalues and the gram matrix for ...
CommunityBot
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views
Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
- 183
2
votes
2
answers
548
views
Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors
Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
CommunityBot
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vote
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Markov Chain that maximises the entropy creation rate
I am working on MERW (Maximal entropy random walk) for a project.
I want to show that given a graph G, there is $\textbf{only one}$ aperiodic markov chain on G that maximises the entropy creation rate ...
- 11
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votes
28
answers
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Probabilistic proofs of analytic facts
What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
- 91.8k
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0
answers
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Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
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1
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628
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A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
CommunityBot
- 1
3
votes
2
answers
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views
Maximizing expectation of gaussian process over covariance matrix with fixed trace
Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded.
For each $\Sigma \...
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