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.
9,025 questions
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Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
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How should one generate a random set of mappings?
My motivation for this question comes from the study of synchronizing automata. There is a general consensus that random automata are synchronizing and have short synchronizing words. I am hoping ...
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Generating a group by randomly sampling generators
Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
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What is the tropical Robinson-Schensted-Knuth correspondence?
And what are it's applications? A conceptual explanation would be great! Is there an expository note about this somewhere?
Some references have already appeared in the answers and comments below. To ...
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Monte Carlo sampling high dimensions with the halton sequence?
Referring to the Halton Sequence, Swiler et al 2006 state that
In cases where a large number of input variables are sampled,
Robinson and Atcitty recommend using a leaped sequence, where the
...
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Generalization of Gauss's inequality for not necessarily unimodal distributions?
Gauss's inequality is for unimodal distributions, concerning distance from the mode.
A similar result is Vysochanskiï–Petunin inequality, which is for the distance from the mean rather than the mode....
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span of symmetrically truncated symmetric random variables
If $X_i$ are symmetric independent random variables, is $\vert \sum X_i I_{\vert X_i \vert < N_i}\vert $ stochastically smaller than $\vert \sum X_i \vert$ ? Is it comparable in any way which ...
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Cover time and intersection time of random walks
Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p_{ij}=1/(2d(i))$ where $d(i)$ is the ...
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Stochastic processes having Markov kernels
Let $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$ be probability spaces and suppose $(X_t)$ and $(Y_t)$ are real-valued stochastic processes defined on the respective spaces. ...
- 1,122
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Brownian motion & time shift
Hi,
I am just starting to study the theory of Brownian motion and I was wondering whether the following was true.
We consider a one-dimensional, one sided, Brownian motion process.
For $A$ an ...
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Concentration of functions of random unitary matrices
Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$...
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Divisible Random Variables
Suppose I can write a positive, real valued random variable
$$ X = m_1 X_1 + m_2 X_2,$$
where $m_1$ and $m_2$ are i.i.d, $X_1$ and $X_2$ are i.i.d and moreover, the $X_i$ are distributed like $X$. ...
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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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Expected value of a logarithm of a Levy process
I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so $\...
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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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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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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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Rank $k$ of a sequence of random variables
Suppose one has $n$ real random variables $X_1, X_2, \dots, X_n$ from a certain distribution. Sort these random variables to get a sequence $Y_1, Y_2, \dots, Y_n$. What is known about the distribution,...
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Modelling a GI/G/1 queue with exceptional first vacation and multiple vacations
I am currently working on a stochastic modelling problem in networks. I have a G/G/1 queue where
the interarrival and inter-service times are iid
the arrival and service time distributions are ...
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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 $$\lim_{n\to\...
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Bounds on tails with moments
A sort of continuation of Comparing distributions with moments
Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\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 ...
- 1,648
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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 ...
CommunityBot
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752
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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
$$s_m(w)= \int_{\bR^m} w(|x|) dx,\;\; ...
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effective/constructive/algorithmic probability theory
What sort of "alternative" probability theories are out there in which the methods of proof are inherently constructive?
I know of a number of theorems that say that if you take an infinite sequence ...
- 2,305
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Probability of a Random Walk crossing an increasing function of the standard deviation
Let $(S_n)_{n=0}^{\infty}$ be a random walk with $S_n = \sum_{i=1}^n X_i$, and let the $X_i$ be distributed according to some (bounded) distribution function $F$ with mean $0$ and variance $1$, so ...
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A random walk with uniformly distributed steps
The following problem has bothered me for a long time.
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly in the ...
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Laplace transform on the cone of positive-definite matrices
The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices.
One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for ...
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Generating Conditional Random Graphs
Let $G(n,p)$ be the usual random graph on $n$ vertices with each edge existing independently with probability $p$ (no self loops , or double edges not are allowed). I would like to simulate the ...
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A uniqueness proposition involving Erf, the error function
This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function".
Consider the system of equations:
$$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...
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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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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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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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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 "...
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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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A generalization of the Sanov Theorem
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with law $\mu$. The Sanov Theorem then states that the empirical measures
$$
\mu^N =\frac{1}{N} \sum _{n=1}^N\delta _{X_n}
$$
...
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Exchangeable normal distribution mixing measure
I have a zero mean multivariate normal probability distribution where WLOG each marginal variance is unity and all pairwise correlation coefficient are equal and positive. The number of elements in ...
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probability of IID sum being positive
Let $X_1,X_2,...$ be iid random variable with mean zero. If $X_1$ has second moment then by the CLT we have $P(X_1+X_2+...+X_n\geq 0)\rightarrow \frac{1}{2}$, as $n$ goes to infinity. I am curious ...
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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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Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes
This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly ...
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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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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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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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Help with derivation of probability density of {event generation} & {event detection}
I would like to specify a new probability distribution that relates to an event of size M being produced by some process and subsequently detected.
Some assumptions :
1) If the event is detected ...
CommunityBot
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A simple stopping time problem.
This should be rather standard so I hope somebody with a good background in probability theory would give me a quick solution or a reference.
We are given a threshold positive integer $T>0$. Let $...
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What is the maximum diameter of $N$ steps of a random walk?
Since probability is quite far away from my daily buisiness, please forgive me if my use of terminology is wrong or the question is too trivial. However, I was not able to find the right keyword to ...
- 413
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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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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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Polynomials on the Unit Circle
I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...
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