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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Graph distance of close points within the minimum spanning tree
My question is the following: Given $N$ uniform IID points $X=(X_1,...,X_N)$ on the unit cube of $\mathbb{R}^d$, take $X_1$ and another point, say $X_{(1)}$, "close" to $X_1$ (i.e. connected to $X_1$ ...
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Probability of event occurring before either of two stopping conditions
Overall problem: Sample i.u.d. from $\{1,\dots,n\}$. What is (a good lower bound for) the probability of getting the values $1$ and $2$ before either you get a number you have seen before or you have ...
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Expectation of the trace of an inverse of a random matrix
Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. ...
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When is an ODE a good approximation to an SDE?
Suppose $X_t$ is a weak solution to a stochastic differential equation in the form
$$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$
for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R^...
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Existence of limit measure
Let $X$ be a separable metric space, $\mu_{n}$ a sequence of Borel probability measures
and $\mathcal{C}$ be a family of sets that is closed under finite unions and
interections, and that contains all ...
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Simulating random sequential adsorption in reverse
Please consider two processes:
Process 1 - I simulate random sequential adsorption of discs on the unit square in the continuum limit, randomly selecting real number coordinates and rejecting the ...
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Obtaining conditional probabilities as pushforwards of [0,1]
It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable ...
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Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables
Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& (\frac{1}{...
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What is the characteristic function of the devil’s staircase?
Let the distribution function $CDF(X,t)$ of a random variable $X$ be defined as $0$ for $ t <0, \text{Cantor function}(t)$ for $t \ge 0$ and $ t \le 1, 1$ for $ t > 1$ (for example, see http://...
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Strictly positive definite autocovariance function of fGn
Hi,
let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter.
I want to show that $\gamma$ is ...
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A simplified MCMC / MH algorithm. Are there known convergence results?
Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
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what is the cycle length of the maximum normalized cycle in the directed complete graph?
Consider the complete, directed graph on $n$ vertices. Let the edge lengths $\{X_{ij}: 1 \leq i, j \leq n\}$ be i.i.d standard normal, with the constraint $X_{ij} = -X_{ji}$. The value of a normalized ...
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1
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Empirical distribution of a collection of iid Markov chains
Suppose we have $N$ independent 2-point Markov chains each having a rate matrix $Q = [-1,1;1,-1]$ and stationary distribution $\pi = [0.5,0.5]$. At time $t=0$, we initiate the chains so that the ...
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Examples of Slowly Mixing Chains in Statistics
This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
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Product of two sigma fields
Let $\mathcal{F}$ and $\mathcal{G}$ be any two famillies of subset of a space $X$ (neither $\mathcal{F}$, nor $\mathcal{G}$ is a sigma-field).
$$\sigma( A\times B , A\in \mathcal{F}, B\in \mathcal{G})...
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Chernoff bound in the not-quite-sub-exponential case
In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables:
http://terrytao.wordpress.com/2010/01/03/254a-...
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Is this probabilistic principle for stochastic processes known?
In the course of a proof, I used the following principle, which seems so intuitive that it should have a name:
Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) ...
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1
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Nonstandard definition for the generator of a standard Ito diffusion
For a standard Brownian motion, the generator of the diffusion is
$$
L = \frac12 \frac{d^2}{dx^2}.
$$
Is there a nonstandard definition of this generator?
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What is the optimal distance to walk back and forth if you don't know how far away or which side the target is on?
Suppose you are facing an infinitely-long wall. Somewhere in the wall is a door, but you can only see the door if you are right next to it. You want to go through the door.
You don't know whether ...
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1
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"Uniqueness of extension" results for measures on separable spaces
Hello all.
I have the following (perhaps basic) question: Let $X$ be a separable metric space. Does there necessarily exist a countable set $\mathcal{C}$ of Borel sets in $X$ such that any two ...
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Intuition behind the spectral density of random matrices
Hi,
I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on ...
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Random variables invariant under almost automorphisms.
Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...
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Expected period of quadratic generator
I am interested in the mean period of a quadratic congruential generator. Let $X_{n+1} = \sum_{i=0}^2 a_i X_n^i \bmod m$ where the $a_i \in \mathbb{Z_m}$ are chosen uniformly at random and $m$ is a ...
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Master theorem for probabilistically inspired recurrences
Is there a general solution for multi-variable recurrences of the following form which come from Markov chain analysis?
$f(i,j,k) = p_1 f(i-1, j,k) + p_2 f(i, j-1,k) + p_3 f(i,j,k-1) + p_4 f(i-1,j-1,...
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How to prove ergodic property from aperiodicity and positive recurrence
How to prove that in case of an irreducible, aperiodic and positive recurrent Markov Chain time average along sample paths is equal to the ensemble average ? i.e.
$$\lim_{n\to \infty }\frac{1}{n}\...
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A probability problem
In my research (I am not a mathematician by training, but I frequently use mathematics in my research), I often come across "approximation" problems in the following form:
Let $X$ and $Y$ be two ...
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Two different definitions of Erdos-Rényi random graph
There are two or more ways to define an Erdos-Rényi random graph. Let consider the following two:
1) $G_n=(V_n,E_n)$ with vertex set $V_n=(1,\dots,n)$ and edge set $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \...
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The property of a Markov measure
Given $\sigma$ a shift map, $m$ - a Markov measure, $C_a$, $C_b$ - cylinder sets.
Suppose $P \in C_b$. The problem is to show the following
\begin{equation}
m(C_a \cap \sigma^{-1}(P)) = \frac{m(C_a \...
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Independent bond percolation on upper density zero subgraphs of the square lattice can have a non-trivial critical point ?
Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$.
Let $V\subset\...
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Is every bornological space measurable?
Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure ...
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1
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Nontrivial lower bounds on Cheeger inequalities for Markov chains
For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when ...
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Expected value as decision criterion in the context of rare events
I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a ...
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compute the waiting time for a given pattern with Kac's lemma
Suppose we are tossing a banlanced coin, and we want to compute the expectation of the waiting time for the pattern HTHTH.
Kac's lemma is a result in ergodic theory which states that, for a ergodic ...
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1
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KL divergence(s) comparison,
Hi,
$P_1$, $P_2$, $P_3$ are probability distributions defined on the same support.
Knowing that $H(P_1) < H(P_2) < H(P_3)$, can we compare $D_{KL}(P_2,P_1)$ and $D_{KL}(P_3,P_1)$ ?
(H is the ...
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Have you seen this one parameter family of distributions before?
This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
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What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?
There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \...
CommunityBot
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When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?
Let $E$ be a locally compact Hausdorff space with countable base and $X_t$ be a stochastic process taking values in the one-point compactification of $E$ (with the Borel $\sigma$-algebra). Let $f$ be ...
CommunityBot
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Joint (close to uniform) distribution in finite fields
This is perhaps a simple fact but I am struggling to prove it.
If A, B are distributed over some finite field $\mathbb{F}$, such that $aA + bB$ is $\epsilon$-close to uniform in $\mathbb{F}$ for ...
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mathematical expectation of length of dependency well.
We have these assumptions:
$W$ is a finite set
$\mathcal W$ is the set of all functions $f:W\to \mathcal P(W)$.
$p:\mathcal P(\mathcal W)\to [0,1]$ is a probability measure.
For each $w\in W$ and $m\...
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Infimum of the Dirichlet form for a tensor product
If $Q$ is the generator of a well-behaved continuous-time Markov process on a finite state space and $p$ is the invariant distribution, the corresponding Dirichlet form is $\mathcal{D}_Q(f) := \frac{1}...
CommunityBot
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Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
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Distribution of the spectrum of large non-negative matrices
This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.
When considering (entrywise) ...
CommunityBot
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
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1
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Chernoff-Hoeffding bound for complex values
Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value
$\mu$ and satisfying $|X_i| \le b$.
Let $\epsilon > 0$. ...
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Good books on stochastic partial differential equations?
I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are welcome....
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Exponential (or other) families of distributions on manifolds.
The exponential family is a general parametrized class of probability distributions on $R^n$ that has many nice properties (ML estimation among them) and includes most of the "standard" distributions ...
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question about circular law
Hi,
I have a question about the circular law.
Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ ...
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1
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773
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Sub-exponential tail implies Poincare inequality
Assume we have a probability measure $\mu$ on $\mathbb R^n$. Assume it satisfies
$$
\mu(||x|| > u) \le Ce^{-au} \ \ \forall u > 0
$$
In other words, its tail is dominated by an exponential ...
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4
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2
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392
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Book on the Moment Problem
Is there a recently published book on the Classical Moment Problems and related theory?
I have seen a couple of old books by Tamarkin and a few other books by Russian authors. Want to know what else ...
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Constructing black noise with non-standard analysis
With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...
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