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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A M/M/$\infty$ queue of depositors with compound interest
Hello, I'm trying to model a bank's liabilities using a queue. Suppose a bank begins with a cash reserve of $M$. Depositors are a M/M/$\infty$ queue; they arrive with rate $\lambda$ and deposit 1 ...
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Is there a regular Dirichlet form with no associated Feller process?
I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process ...
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Fourier transform of distributions with non-standard test functions
This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:
Given a ...
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A Pascal's-triangle -like random process
I was exploring Pascal's triangle on a cylinder when I encountered this puzzle-like problem.
It is surely elementary, but perhaps weekend-entertaining.
Start with a permutation of $(1,2,3, \ldots, n)$...
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Probability inequalities
Hi everyone,
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an ...
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Renewal process: domination by product measure
Consider a stationary process $(X(i), i\in\mathbb{Z})\in \{0,1 \}^\mathbb{Z}$ with the following structure; runs of 0s alternate with runs of 1s, with the length of all runs independent, and with each ...
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Average Hamming distance between strings after some number of random substitutions in a population of initially identical elements
Let's say I have a set $S$, $(s_1, ..., s_i, ..., s_P) \in S$, of $P$ identical strings over a $k$-letter alphabet, each of length $|s_i| = L$. With uniform random probability across all strings in $...
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Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
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700
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convergence in distribution of stochastic gradient descent.
The stochastic gradient descent algorithm where only a noisy gradient (zero mean noise) is used to update current estimate is known to converge almost surely to the minimizer. However, if one is ...
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To what extent can the following zero-one laws be relaxed?
I am interested in what circumstances various zero-one laws in probability theory can be relaxed. In particular, independence is a very important factor in such laws.
1) Borel-Cantelli Lemma: Let $...
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Correcting bias in samples selected by a prediction
Here is the scenario:
I'm trying to find as many golden tickets as I can, so that I can sell them to kids that want to go on a tour of Wonka's chocolate factory.
Fortunately, I have a machine that ...
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A question on random walks on semisimple groups
Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For ...
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Green function of simple random walk
Let $G$ be the Green function of the simple random walk on $\mathbb{Z}^d,\:d\geq 3$; i.e.
$$G(x) = E \sum_{i=0}^{+\infty} 1_{X_i=x},$$
where $X$ is the simple random walk starting from $0$. The ...
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Alternative approaches to probability theory
I'm undergraduate student in probability theory (and its applications). There are lots of different and definitely good text on standard, functional analysis-based approach, but I'm interested in ...
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What results would follow from or imply "randomness" of the primes?
This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
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Random versions of deterministic problems
What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version?
An example ...
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Uniform correlation matrix sampling and not so uniform laws
Hi everyone,
I am looking for a way of simulating correlation matrices of fixed dimension in (at least) two ways.
First, I would like to determine the "uniform" distribution over the "correlation ...
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Random polycube shapes
I am wondering if it is hopeless to obtain any firm results
on the following model of a "random polycube shape."
First, a polycube in $\mathbb{R}^3$
is a connected face-to-face gluing of unit cubes.
(...
- 151k
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Intrinsically measurable subsets of amenable semigroups.
This question is related to the one in https://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be ...
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Are these two definitions of "uniformly distributed" equivalent?
For an article I am writing, I would like to know that two somewhat different
looking conditions are in fact equivalent. Here is the setting. $X$ is a compact
(and first countable) metric space and $\...
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Solution to difference differential equation with constant coefficients
This problem arose when solving a continuous Markov chain exercise from a book I am studying. Given a set of positive $q_i$ with $i \in \mathbb{Z} $, and non-negative $\lambda$ and $\mu$ that add to 1,...
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Is there some generalization of the "Maximum Coverage Problem" for information in random variables?
Say I have a set $X=x_1,x_2,\ldots,x_n$ of random variables, and would like to find a size $k\leq|X|$ subset that contains as much information as possible. This is complicated because the variables ...
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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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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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Special case of Duffin-Schaeffer conjecture
The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: http://en.wikipedia.org/wiki/...
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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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421
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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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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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Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
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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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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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Multiplying two probability distributions represented by particles
Hello,
I have two probability distributions p1(x) and p2(x) given by (x_1i, w_1i) and (x_2i, w_2i) respectively, i.e. they are both represented by sets of particles. I need to create the pdf p(x) = ...
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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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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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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( \...
- 1,068
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Analytical expression for variance of nested binomials?
Hi all,
I want to compute the variance of a variable that is defined at each step as a recursion of binomials in the following way:
A=1
B=Bin(1,A)*Bin(1,p)
C=Bin(1,B)*Bin(1,p)
D=Bin(1,C)*Bin(1,p),...
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Probability of a Random Walk crossing a straight line
Let $(S_n)_{n=1}^{\infty}$ be a standard random walk with $S_n = \sum_{i=1}^n X_i$ and $\mathbb{P}(X_i = \pm 1) = \frac{1}{2}$. Let $\alpha \in \mathbb{R}$ be some constant. I would like to know the ...
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Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...
- 151k
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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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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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What can be said about an infinite linear chain of conjugate prior distributions?
We can sample a discrete value from the multinomial distribution.
We can also sample the parameters of the multinomial distribution from its conjugate prior the dirichlet distribution.
Since the ...
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Maximum of Gaussian Random Variables
Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.
Let $m$ be the maximum of the random variables $x_{i}$
$$
m=\max\{x_i:i=...
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Textbooks to use as reference for standard calculus and probability topics
I am currently working on a paper to be submitted to a US journal (addressed primarily to non-mathematicians’ audience) where I use
(1) some standard calculus stuff (e.g. limits, Taylor expansions, ...
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Marginal distribution of the diagonal of an inverse Wishart distributed matrix
This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here:
Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the ...
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Question on randomness extractors
Person A has a source $W$ with min-entropy($W$) = $k$. He also has an extra piece of information about the random source, denoted with $y$, such that min-entropy($W|y$) = $k/3$.
The adversary doesn't ...
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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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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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Proof of conditional copula relation to the marginal copulas
Hello
I am trying to derive the second equation displayed in section 7.1 (or p. 41) of this article or equation (6.3) of this book. I've seen this in many documents discussing conditional sampling ...
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Packing density of randomly deposited circles on a plane
Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$:
Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this surface,...
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