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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Randomized algorithm?
The problem is as follows. Given a set $S$ of natural numbers of size $n$ where each $x_i \in S$ is from the set $[n^2]$. Elements of $S$ are not necessarily pairwise different, i.e., there can be ...
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Distribution of the standard deviation of normal variates
What is the distribution of the standard deviation of $n$ normal variates? That is, if $X_1,...,X_n$ are i.i.d. normal random variables with mean $\mu$ and s.d. $\sigma$ and $M=\sum X_i/n$, then what ...
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Conditioning on one term of a sum of random variables
Let $\theta$ be normally distributed with mean $\bar \theta$ and variance $s^2$. Let $Z$ be normally distributed with mean $0$ and variance $\sigma^2$, and chosen independently of $\theta$. Define $...
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What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?
This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
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Joint law of the time integral of Brownian motion and its maximum
Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\...
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some weird relations among beta random variables
Let $X_i, Y_i, Z_i$ be three iid family of standard normal random variables. Then the following random variable is distributed the same as the first coordinate of a uniform $2$-sphere:
$$ \frac{X_1}{\...
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Given a Levy Exponent find the jump-measure and drift
A Levy subordinator is an finite variation Levy process with non-negative drift and positive jumps. The Levy exponent is given by
$$\phi(\lambda) = \gamma \lambda + \int_0^\infty ( 1 - e^{-\lambda s}...
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Law of large numbers for stochastically chosen samples
Let $X_t$ be a sequence of i.i.d. random variables with mean $\mu$. Then the law of large numbers states that
$$\lim_{T \to \infty} \frac1T \sum_{t=1}^T X_t = \mu \quad a.s.$$
Now suppose that (in a ...
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Does a definition for delta sequences in the multidimensional case exist?
does anybody know a good book on multidimensional delta sequences?
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CLT for the squares of unbounded non-identically independently distributed random variables
I have a sequence of independent but not identically-distributed random variables $X_1, X_2, \ldots, X_n$ where $X_i\sim A_i$, with each $A_i$ having a support over $\mathbb{R}$ and subject to the ...
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Terrain Generation: Infinite 2D space filled with Diffusion-limited aggregation clusters?
Disclaimer: I don't have a deep understanding of fractals or any higher math, I'm just personally interested in it, so please excuse me if I'm using wrong terms or if I'm being inaccurate. Making ...
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Doubly stochastic matrices as squares of entires of unitary matrices
Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a ...
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Expected norm of sum of random orthogonal matrices
Somehow I got wondering about the following question today:
Suppose $Q_1,\ldots,Q_n$ are random (uniformly sampled) $d \times d$ orthogonal matrices.
What is the expected value of the quantity $\|\...
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Stable Law with Rates
If $X_{i}$ are a bunch of iid random variables with mean 0 and finite second moments, we know that $\sum_{i=1}^{n} \frac{X_{i}}{\sqrt{n}}$ converges in law to a Gaussian. Furthermore, by the Berry-...
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Is the Hausdorff metric on sub-$\sigma$-fields separable?
Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
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Random distance matrices
My question is motivated by the following recent paper:
Gadgil, Siddhartha; Krishnapur, Manjunath, Lipschitz correspondence between metric measure spaces and random distance matrices, Int. Math. Res. ...
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proofs of stochastic boundedness
I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.
In particular, I'm trying to ...
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Is there a fair coin?
I attended a course on stochastic processes a few years ago. During the course the lecturer mentioned that there is a mathematical proof (with some assumptions, naturally) of non-existence of a fair ...
user10891
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Convergence of stopped Brownian motion
Suppose $B$, $B_n$ are Brownian motions, and write $B^s$ for $B$ stopped at the first time equals $k$, say. (Similarly $B^s_n$).
I know how to prove the following: if $B_n \to B$ uniformly on ...
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How similar are discrete stable RVs to their continuous analogues?
The generalized central limit theorem of Gnedenko-Levy describes the asymptotic behavior of a sum of IIDRVs which may not have finite mean or variance. Only a small class of limit laws can be realized,...
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Length of the last edge when visiting points by nearest neighbor order
Take $n$ points uniformly in $[0,1] \times [0,1]$. Then pick uniformly $X_0$ one of these points as your starting point. Then let $X_1$ be the nearest neighbor of $X_0$, let $X_2$ be the nearest ...
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Exponential (or other) families of distributions on manifolds.
The exponential family is a general parametrized class of probability distributions on $R^n$ that has many nice properties (ML estimation among them) and includes most of the "standard" distributions ...
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Some questions concerning a random number process
Consider the following Markov process: Start with an integer $N = N_0$. Now repeatedly choose an $N_i$ uniformly at random in the range $[1...N_{i-1}]$ until $N_i = 1$ at which point one terminates ...
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Intuitive understanding of the Stieltjes transform
I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does.
The gist of my work is that I have an $N\times N$ true covariance ...
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Overall covariance of Mixture of Gaussian
I have a Mixture of Gaussians to model an arbitrary distribution. I would like to model a distribution derived from this GMM with:
Mean = Weighted average mean of GMM means.
I am not sure about how ...
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Brownian particle with jump boundary condition
I would like to find a function $f(s)$, which solves the following equation:
$ \int_0^t \int_0^L f(s,x) p(t-s,x,y) dy ds = 1 $
The function $p(\tau,x,y)$ is
$p(\tau,x,y) = \sum_n e^{-\lambda_n \tau}...
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Low degree polynomial approximation for the entropy function
Let $X$ be a discrete random variable with possible values
$\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of
$X$. In addition, denote $p_i=p(x_i)$.
The entropy of $X$ is ...
user17253
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Processes approximating a reflected brownian motion.
Let $W$ be a standard Brownian Motion. Let $\epsilon>0$ be given. Let $X^\epsilon$ be the process which diffuses like $W$ on $(-\epsilon,\infty)$, but when $X^\epsilon$ reaches the level $-\...
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Distribution of a maximum
I am reposting a question on math.stackexchange which did not recieve good questions.
The orginal questio is at https://math.stackexchange.com/questions/73091/distribution-of-a-maximum.
Randomly ...
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Probability-one event for Markov chain
Let $X$ be a Markov chain, with countable state space $I$ and transition probability matrix $P$. $X$ is irreducible, but need not be recurrent. Let $S$ be a fixed subset of $I$.
Define a subset $K$ ...
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For what range of edge probability does the following property hold for random graphs?
Let $G(n,p)$ denote the Erdős–Rényi model of random graph. For a given function $p = p(n)$ we say that $G \in G(n,p)$ asymptotically almost surely has property $\mathcal{P}$ if
$$\mbox{Pr}[G \mbox{ ...
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Bounding the success time of a coupon collector like problem
Consider the complete graph on $n$ vertices. Each step, one chooses one of the $\binom{n}{2}$ edges iid uniformly at random. Say a sequence of choice is successful if there is some permutation of the ...
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Lower bounds for partial sums of multiplicative functions
The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
$$F(s)=\...
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Markov Property: determined by just the law or also the realization?
When one says that a stochastic process is Markovian, is this a property solely of the law of the process, or does the realization of the process also come in to play? I am asking even for the ...
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Copulas and time series
Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?
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Elementary analysis: reference request
Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:
$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$
So essentially ...
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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 ...
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Absolute moments of symmetrical distributions
Suppose $F~$ is a probability distribution symmetrical about 0, for which all moments exist. Let $\mu_i~$be the $i$-th moment (of course $\mu_i=0$ if $i~$ is odd).
We know there are some conditions ...
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Multidimensional Berry–Esseen for probability density functions
This is a follow up to this recent question: Berry Esseen type result for probability density functions
There exists a multidimensional version of the usual Berry–Esseen theorem (for cumulative ...
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Normal distribution with positive SEMI-definite covariance matrix
In [1] is noted, that a covariance matrix is "positive- semi definite and symmetric". I wonder if it is possible to a multivariate normal distribution with a covariance matrix that is only ...
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Iterated Kumaraswamy distributions
The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$.
Does anyone know any formulas or properties relating to iterations of this on itself, meaning
$$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$
If ...
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Eigenvalue density of some random matrices?
Consider a class of real symmetric random matrices,$M_{n\times n}=(X_{i,j})_{n\times n},$ whose off-diagonal elements follow an exchangeable distribution, and the diagonal elements follow another ...
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Random Reidemeister moves to unknot
Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing ...
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Topple height of randomly stacked bricks
What is the expected height of a stack of unit-length bricks, each one
stacked on the previous with a uniformly random shift within $\pm \delta$?
The stack topples if the center of gravity of the top $...
- 151k
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Statistics for Haar measure of random matrices?
Let's say I have $M$ samples of $N\times N$ real orthogonal matrices. What statistics can I calculate to test if they could have been drawn from a distribution consistent with Haar measure over $O(N)$?...
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Conditional prob of value given input is drawn from subset = conditioning over subset?
Hey guys, I have a pretty basic question that I want to be sure of. I'm taking a probability over an input selected uniformly at random from binary strings of length $l(n)$. I would like to compare ...
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Eigenvalue densities of sample covariance matrices when the population covariance matrix is a perturbed identity matrix
TLDR: I'm looking for a random matrix theory reference for the eigenvalue densities of sample covariance matrices (both dimensions approaching infinity at the same rate) when the true (population) ...
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Distances between and among points in a region
Let $X = \{x_1, \dots, x_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum_{i=1}^n \| x_i \| $ and let $G(X) = \iint_S \min_i \|x - ...
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How are epidemic models simulated in case of mobility?
I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
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Sequences of linear combinations of measures
Let $X$ be a Polish space. Let $J\in\mathbb{N}$.
Let $\lbrace a^n_1\rbrace_n,\dots,\lbrace a^n_J\rbrace_n$ be $J$ sequences of reals.
Let $\lbrace \mu^n_1\rbrace_n,\dots,\lbrace \mu^n_J\rbrace_n$ be ...
user12713