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.
9,027 questions
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Random Permutation with fixed cycle length.
Suppose $ S_{n,N} $ be the set of $n$ elements with $N$ many cycles where $N$ is proportional to $n$. $U_{n,N}$ is an element picked randomly from this. It is known that the length of any cycle cannot ...
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How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
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Multivariate power law distributions?
Is there a text books or publications that describes multivariate power law/pareto distributions?
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Convergence of sample mean
I have a two-index succession of real-valued random variables $x_{t,n}$ such that $\lim_{n\to\infty} x_{t,n} = x_t$, for all $t$ and suitable limit r.v. $x_t$.
I would like to prove that $$\lim_{n\to\...
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resampling over Bowen balls
Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...
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Bounds on tails with moments
A sort of continuation of Comparing distributions with moments
Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\...
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Entropy of Bernoulli walks on semi-groups.
Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...
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Intersection of an uncountable number of sets.
Let $\mathcal{I}$ be an uncountable set. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, and $E_i, i\in \mathcal{I}$ be a measurable set such that $\mathbb{P}(E_i)=1$. What can we say ...
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Comparing distributions with moments
Suppose I have two variables $X$ and $Y$ which have continuous p.d.f.s $f$ and $g$ on the positive real line. I know that the moments $\mathrm{E}[X^n] > \mathrm{E}[Y^n]$ for sufficiently large $n$ (...
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karhunen-Loeve expansion of Poisson process
Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda t$...
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Number of connected components in a graph from G(n,m)
Hello,
$G(n,m)$ is the family of all graphs with $n$ vertices and $m$ edges (I consider $m < n$).
Each graph in $G(n,m)$ is selected with uniform probability.
What is the probability that the ...
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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}...
- 1,256
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Probability of a Random Walk crossing an increasing function of the standard deviation
Let $(S_n)_{n=0}^{\infty}$ be a random walk with $S_n = \sum_{i=1}^n X_i$, and let the $X_i$ be distributed according to some (bounded) distribution function $F$ with mean $0$ and variance $1$, so ...
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"psi-epistemic theories" in 3 or more dimensions
In their recent paper The Quantum State Can Be Interpreted Statistically, Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for ...
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multivariate Gaussian approximation in total variation distance
I'm wondering if there's any general technique that gives the total variation distance between a distribution on $\mathbb{R}^n$ and $N(0, I_n)$.
My understanding is that Stein's method gives only ...
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Generating Conditional Random Graphs
Let $G(n,p)$ be the usual random graph on $n$ vertices with each edge existing independently with probability $p$ (no self loops , or double edges not are allowed). I would like to simulate the ...
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Time reversibility of Stratonovich Diffusion: Reference Request
Please consider the Stratonovich stochastic differential equation (SDE)
$$
dX = b(X)\circ dB
$$
where $B$ is standard Brownian motion and $X(0)=X_0$. This corresponds to the Ito (SDE)
$$
dX = \frac{1}{...
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If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense
Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense ...
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A uniqueness proposition involving Erf, the error function
This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function".
Consider the system of equations:
$$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...
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Inequalities involving moments
$\newcommand{\bR}{\mathbb{R}}$ Suppose that $w:\bR\to \bR$ is a nonnegative, even smooth function decaying fast at $\infty$, $w\in\mathscr{S}(\bR)$.
Define
$$s_m(w)= \int_{\bR^m} w(|x|) dx,\;\; ...
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Dither in Leech lattice quantization!
Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the Voronoi region of the Leech lattice.
Best,...
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Anti-concentration of Gaussian quadratic form
Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. My question is: Do there exist absolute constants $C,c>0$ such that for every $\epsilon>0$ and positive real numbers $a_1,\...
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On randomly colored random chords
Let $Q=[-1/2,1/2]^2$ be a unit square and let $(\ell_n,\varepsilon_n)_{n\geq1}$ be an iid sequence of isotropic lines intersecting $Q$ (more precisely, distributed according to a Haar measure on the ...
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"Graphical models" and "gene finding and diagnosis of diseases" ?
Quote Wikipedia: Applications of graphical models include ... gene finding and diagnosis of diseases...
Unfortunately there is no comment what are these applications...
Can one comment on this ?
...
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Central limit theorem for 3d rotations
Let $X_i$ (where $i=1, \dots , n$) be independent and identically distributed 3d rotations. What is the distribution of $X_1X_2\dotsb X_n$ in the limit of large $n$?
I'm especially interested in the ...
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A random walk with uniformly distributed steps II
The problem is a improved version of this problem,
A random walk with uniformly distributed steps
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "...
- 1,551
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Laplace transform on the cone of positive-definite matrices
The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices.
One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for ...
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Transience of self avoiding random walks on $\mathbb{Z}^d$
I'm finishing up a masters thesis in computer science and want to say a bit in the introduction about self-avoiding walks. My thesis looks at a random process which arose in computer science and my ...
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Limit of a rescaled random sum of i.i.d. random variables
Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
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Reducing system of equations involving Erf, Error Function
I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ...
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How to bound the sup norm of a Rademacher process or equivalently a Gaussian process?
I want to know how to find an upper bound of the following expectation taken for both $t$ and $y$ as
$$\mathbb{E}\sup_{x \in D} \left|\sum_{k=1}^n t_k x^T y_k\right|,$$
where $D$ is the set of ...
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Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
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Genericity of sets without unique mean value
Following Rosenblatt and Yang, I say that a subset $A$ of $\mathbb Z$ has a unique mean value if for all invariant means $\lambda_1,\lambda_2$ on $\mathbb Z$, one has $\lambda_1(A)=\lambda_2(A)$.
...
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distribution of specific exponential functional of brownian motion
Does the following hold true $\forall T>0,a>0,c>0$ (in particular for c arbitrarily small):
$P_0(\int_0^T e^{-aB_s}ds<{c})>0$?
This is a minor result which will improve several ...
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A random walk with uniformly distributed steps
The following problem has bothered me for a long time.
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly in the ...
- 1,551
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Capped binomial random variables
Consider a random variable $X = \sum_{i=1}^{m} X_i$, where each $X_i$ is an indicator
random variable that is $1$ with probability $k/m$ and $0$ otherwise. We are interested in the quantity $S_X(m) = ...
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Classical convolution VS Free Convolution
We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi_*:M_1(\mathbb R^2)\rightarrow M_1(\mathbb R)$ the induced push-forward map (where $M_1(\Delta)$ stands ...
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Affect of noise on Random variable separation
We have two random variables $X$ and $Y$. Suppose $P_1$ is the probability that $Pr[X > Y]$. $Z_1$ and $Z_2$ are two i.i.d. (identical and independent) random variables, and let $P_2$ be the ...
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Correlation-Function for Random Graph Ising Model
For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes (...
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James-Stein phenomenon: What does it mean that a James-Stein estimator beats least squares estimator?
Background James-Stein estimator and Stein's phenomenon, as described in Wikipedia are rather counterintuitive and amazing.
It is claimed that if one wants to estimate the mean $\Theta$ of
Gaussian ...
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De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...
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Concentration of Gaussian vectors
If $f: \mathbb{R}^n \to \mathbb{R}$ is a Lipschitz function and $X$ is a standard $n$-dimensional Gaussian vector with $\mathbb{E} f(X) = 0$, then $f(X)$ is subgaussian (in a way that does not depend ...
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Branching process question
(Cross-posted to math stackexchange question 130154)
I am trying to analyze the following branching process. We start with a root (level 0) node. Each surviving node has two children, each of which ...
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Binomial moments for the number of events occuring
Let $A_1,A_2,\dots,A_n$ be events on a probability space. For $0 \leq k \leq n$ let
\begin{equation*}
S_k=\sum_{1 \le {i_1}<{i_2}<\cdots<{i_k} \leq n} P(A_{i_1} \cap \cdots \cap A_{i_k}).
\...
- 3,142
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Prokhorov theorem
Hi there. It is known that on a polish space, if a family of bounded positive measures (no need to be probabilities) is tight, then it is relatively compact in the space of positive measures with ...
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Forms of the Levy-Khintchine formula
I'm writing a survey that involves Levy processes and wanted to mention the different forms of the Levy-Khintchine formula found in literature.
The most common version seems to give the Levy symbol ...
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418
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"birds on wire" type problem
Consider $n$ individuals {$1,2, \ldots, n$}. For each (unordered) pair of individuals $i \neq j$ we consider a random variable $X_{i,j}$ that can be thought of as the distance between $i$ and $j$. ...
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Are there lightweight foundations for arbitrarily extendable objects?
My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...
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Joint distribution of Ito integral and its quadratic varation
Any idea on solving the joint distribution of
$X_T=\int_0^T \alpha_t dZ_t$ and $Y_T=\int_0^T \alpha_t^2 dt$ ? Here $X_T$ is an Ito integral and $Z_t$ is a standard Brownian process. When $\alpha_t$ ...
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Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?
First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute
$\mathbb{E}[\...
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