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.
9,024 questions
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statistical approach to multinomial distribution
Suppose a dice with $q$ faces is rolled $N$ times, where $N$ is very big.
We define a multinomial variable $X=(X_1,\ldots,X_q)$ which counts how many times any face is occurred ($X_i$ is the number ...
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(almost) statistical independence of nodes degrees in a graph
Wireless networks are typically modeled as random geometric graphs. The number of nodes $N$ in the network is drawn from a Poisson distribution with intensity $\lambda$
$$P(N = n) = \frac{\lambda^n ...
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Fourier Transform of measure on Banach Space (a question about Pontryagin Duality)
The following definition is given as the Fourier transform of a Borel probability measure $\mu$ on $E$, a Banach Space (Real):
$\hat{\mu}: E^*\rightarrow \mathbb{C}$ defined by
$\hat{\mu}(x^*):=\...
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Limit of an integral involving the normal CDF
Let $\Phi $ denote the standard normal CDF, and $\phi$ the standard normal PDF. Fix $\alpha > 0$.
Let
$$ Z\left( r\right) =r\int_{0}^{\infty } e^{-(r+\alpha )t} \mathbb{E} \left[ \Phi
\left( \...
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Sufficiently random sample
Let $d$ be an integer $\geq 2$, and let $\Omega = \lbrace 0,1 \rbrace^d$, $A \subseteq \lbrace 0,1 \rbrace^2 $ and $i,j$ integers with $1 \leq i < j \leq d$. If we select an element $(x_1,x_2, \...
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Looking for an appealing counterexample in probability
There is a commonly-encountered-but-wrong rule of thumb that says something like
If a probability distribution is positively skewed, its mean is greater than its median.
(You sometimes also see it ...
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Best constant in comparison between Rademacher and gaussian averages?
Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables.
What is the best ...
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Non-integrable ergodic theory
Can anyone help me out with proofs/counterexamples? I'm working on an operator-valued multiplicative ergodic theorem and need what may(?) be a well-known fact. This fact (if true) would help me get ...
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Dynamics of a random "quadratic" directed graph
Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" ...
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"Bridging" uniform and "mass" distributions
Foreword. The original formulation of this problem was inaccurate; chamomille and Didier Piau came up with a simple example which would not solve the problem in its accurate formulation. Sorry for my ...
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Probability Problem Involving e
I thought of the following probability problem, which seems to have an answer of 1/e, and wonder if someone has an idea as to how to prove this.
Suppose a man has a bottle of vitamin pills and wishes ...
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Intersection Probabilities for Random Walk in d>2
I'd like to get asymptotics on the probability that n independent random walks coalesce. Start with n independent walks. As soon as two walks intersect they become one walk and continue evolving as ...
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Ergodicity of Convoluted White Noise
I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...
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Probability measure product space
Let $(X,B,\mu)$ and $(Y,C,\nu)$ be probability spaces, and let $m$ be the product measure. Let $f:X \times Y \rightarrow [0,\infty )$ be a $B \otimes C $ measurable function, $1 < p < \infty$, ...
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Simple functional form for correlated Bernoulli variables
I'm looking for a simple, symmetric multivariate distribution for $N$ Bernoulli variables with the following properties:
Each individual variable takes on values 1 or 0
Fix a subset of $M$ variables. ...
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A sequence of order statistics from an iid sequence
Note: This question was asked in stats.stackexchange.com and math.stackexchange.com, with expired bounties on both sites.
Given a sequence of iid random variables $X_i$ (without loss of generality ...
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Random walk origin return monotinicity
Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...
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Rate of decay of variance for a tensor product Markov process (100 pt bounty for good answer by 1800 EST Fri)
Let $Q$ be the generator of a well-behaved (not necessarily reversible) Markov process $X$ on $[n] = \{1,\dots,n\}$ and let $Q^\otimes = \sum_{m=1}^N I^{\otimes(m-1)} \otimes Q \otimes I^{\otimes(N-m)}...
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Probability of having a bounded ratio of two types of balls in each of 'S' bins after random partitioning of a fixed number of balls
Let's say I have a bag with $A$ red balls, $B$ blue balls, and a total number of balls $N = A + B$. With uniform probability, and sampling without replacement from the $N$ balls, I fill an integer ...
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Random Trigonometric Polynomial
Let $t_{1},t_{2},\ldots, t_{n}$ be i.i.d. real Gaussian random variables of zero mean and variance one. Let $a_{1},a_{2},\ldots, a_{n}$ be positive and fixed real numbers and define the random ...
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Random Walk vs Branching process
1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs
with probability 1/N!. The probability $Pm(N)$...
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Generating Bernoulli Correlated Random Variables with Space Decaying Correlations
Hi,
I have a set of N objects randomly distributed in a 2D physical space. Each object (i) generates a bernoulli random number (0 or 1) based on a marginal probability Pr(xi = 1) = p. These objects a ...
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Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
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Renewal function - duality
Let us consider a random walk $(S_n)_n$. One denotes the instants of records of $-S_n$ by $0=T_0 < T_1 < T_2 \cdots$. Then for all $k$ one sets: $H_k=-S_{T_k}$. Finally, one define $\tau$ as the ...
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when is an element of $M_n(M)$ $\ast$-free from $M_n(\mathbb{C})$ for a $\ast$-non-commutative probability space $M$.
From "Lectures on the combinatorics of free probability" by Nica and Speicher we have a necessary sufficient criteria for an element of $M_n(M)$ being free from $M_n(\mathbb{C})$ for a non-commutative ...
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Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices
I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:
$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ ...
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Number of required trials to sample all possible states of a 'd'-sided loaded die
Let's say that I have a loaded $d$-sided die where the relative probabilities for the die landing on a particular side, $(p_1, ..., p_d)$, are known. How many times must I roll the die to, on average,...
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Intersection probability for 'N' fixed-length rods in one- or two-dimensions
Please consider the case where I have 'N' rods of length L (and width W) placed on a one- or two-dimensional surface with dimensions [0, A] in 1D, and [ [0, A], [0, B] ] in 2D. For the two-...
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Maximums of two correlated Gaussian processes
Hi,
This question is motivated by a statistical genetics model.
Let $(x_1,y_1)$, .., $(x_N,y_N), ... $ be i.i.d. bi-variate Gaussian random variables.
The $x_i,y_i$'s are standard Gaussians, $x_i, ...
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is the variance of a test function of a markov chain always increasing?
Edits: Changed function to eigenfunction. I should have stated the problem with more explicit conditions. Anyways I realized the original formulation is not true, even when one starts at a single ...
CommunityBot
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Points on binary hemispheres of the n-sphere
Let $\mathbb{S}^{n-1}=${$ x\in \mathbb{R}^n| \sum_{k=1}^n x_k^2 =1 $} be the $n-1$ sphere and $n_i\in\mathbb{R}^n$ with components $n_{ij}\in${$-1,1$}$\ \forall\ j=1,2,\dots,n$. There are obviously $2^...
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estimating sample size
Say there is a web service where I can request information about a random item.
For a request each item has an equal chance of being returned.
If I keep requesting items and record the number of ...
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Examples of Lyapunov functions for Markov processes
I am reading about Lyapunov functions for Markov processes, and I am having trouble thinking of examples to keep in mind as I read. If $X_t$ is a continuous-time Markov process with generator $L$, a ...
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Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result.
The background for this problem comes from the composition of Brownian motion and ...
CommunityBot
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probability that a random element of Z/NZ can be written as a subset sum of others
How could one calculate the probability that any element in $\mathbb{Z}/N\mathbb{Z}$ can be written as a subset sum of $n$ random elements in $\mathbb{Z}/N\mathbb{Z}$?
In other words, say I pick $n$...
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Connectivity in random points on a grid using a rope of fixed length.
This problem is a by product of another problem. I would like to restate this problem as a sort of a puzzle.
Suppose we have a $l \times b$ grid. We select $k$ points on the grid randomly and ...
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Wiener Sausages in Riemann Surfaces
Let $M$ be a Riemann surface (or a higher dimensional manifold) and let's assume that it's geodesically complete. Let $W(t)$ be a Brownian motion on the surface accordingly to the manifold's Laplacian ...
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Missing mass estimate
Let $S$ be a finite set with probability distribution $P$. Define the random variable $m_i$ to be the "missing mass" after seeing $i$ iid samples from $S$ under $P$. That is, $m_i$ is the total mass ...
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Will a random walk on [0, inf) tend to infinity? [closed]
Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.
As time goes to infinity, will ...
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Estimates for Symmetric Functions
Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial
$$
p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n
$$
where the $a_i$ are ...
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Talagrand's concentration inequality with limited independence
Is there a version of Talagrand's concentration inequality known when the variables have limited independence. More precisely, Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $1$-Lipschitz convex ...
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The $\sigma > 0$ condition in the Central Limit Theorem
In the version of central limit theorem for strictly stationary but weakly dependent (for instance $\alpha$-mixing with fast decaying mixing coefficient) random variables $X_1, X_2, \cdots$, the ...
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Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?
It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
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Probability estimates for "beans & boxes"
From a discussion with some friends, this apparently easy problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial ...
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Probability of return at step $n$ of a Random walk to its starting vertex
Hi,
given a discrete simple Random walk on a symmetric graph, what is known about the probability of the random walker to return to a starting site at step $n$? Specifically, I am interested in the ...
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How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?
Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise ...
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Assigning positive edge weights to a graph so that the weight incident to each vertex is 1.
Let $\Gamma=(G,E)$ be a connected undirected graph, with no loops or multiple edges. $G$ is finite or countably infinite. For each edge $e=\{x,y\}\in E$, we assign a positive, symmetric edge weight $...
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Good probability measues on $S^1$ reprented by a kernel
I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
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Weierstrass' function and Brownian motion
Is there a known connection between Weierstrass' function
$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$
and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass ...
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Correlation structure among the maximums of a Brownian motion
Is there a known correlation structure among the maximums of a Brownian motion on disjoint intervals ?
Let $(W_t)_{t\geq 0}$ be a one-dimensional standard Brownian motion,
and take the partition $0=...
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