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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Bounds on CDF for the median of samples from an exchangeable distribution
Suppose $x_1,\dotsc, x_n$ are $n = 2k-1$ samples from an EXCHANGEABLE sequence, where the common marginal distribution is assumed UNIFORM on $[0,1]$. Let $x_{(1;n)} \le \dotsc \le x_{(n;n)}$ be the ...
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SOS model - height
Let $\Lambda_n = \{ -n,\ldots,n\}^2$ and let $\{X_i\}_{i\in \Lambda_n}$ be the family of $\mathbb{R}$-valued of random variables with the density proportional to
$\exp(-\sum_{i\sim j} |X_i - X_j|),$
...
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2
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Infima of conditional densities after disintegration
Consider the measurable partition of the open unit square $(0,1)\times(0,1)$ into horizontal intervals $L_y=(0,1)\times\{y\}$. Let $\mu$ be a Borel probability measure with the disintegration
$$
\mu(...
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Traceless GUE : Four Centered Fermions
The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
$$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
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Combinatorial proof for the number of lattice paths that return to the axis only at times that are a multiple of 4
Consider lattice paths consisting of $2n$ steps, each of which is either $(1,1)$ or $(1,-1)$. The number of such lattice paths that return to the horizontal axis only at times that are a multiple of $...
- 3,283
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When are time changes of Feller-Dynkin processes still Feller-Dynkin processes?
A Markov process $X_t$ on $E$ is a Feller-Dynkin (or sometimes just Feller) process if its semigroup is a strongly continuous, sub-Markov semigroup $\{P_t:t\geq 0\}$ of linear operators on $C_0(E)$ (...
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Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 12.8k
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Lower bound on sum of independent random variables
Assume $0 < a_i \leq 1$ for $i = 1, 2 \ldots n$. I am interested in the random sum $X = \sum_i a_i X_i$ where $X_i$ are iid random Bernoulli variables with some mean $p \in (0, 0.5)$. I would like ...
- 501
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Continuous family of Markov chains
Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition ...
- 1,069
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Probability, preferential attachment, "rich get richer"
Imagine you have $N$ empty bins. At every timestep $t$ you throw a ball to a randomly chosen bin ($t$ is therefore also the total number of balls in this system). Probability that a ball falls into a ...
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440
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Stochastic Integration via Skorohod Representation
I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; ...
- 305
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Path integral and harmonic oscillator
Maybe this is not a research level question. I post it because I heard that the path integral can be rigorous by Brownian motion. But my knowledge of probability is so limited.
If $$L=\frac{1}{2}(-\...
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Distribution of uniform-normed random vector
What is the pdf of $\vec{Y} = \frac{\vec{X} }{\lVert \vec{X} \rVert_\infty}$ with $\vec{X}$ a random vector following a multivariate standard normal distribution (zero-mean $\vec{\mu} = 0$ and ...
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$A \perp B$ and $A+B\perp r\left( 2A+B\right)$ for some continuous function $r$. Is there such a triplet $\left( A,B,r\right) $ with non-constant function $r$?
Let $A$ and $B$ be independent continuous random variables with supports $ \left( -\infty ,\infty \right) $ and $r$ be a continuous function. In addition, $A+B$ and $r\left( 2A+B\right)$ are ...
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Use of a priori information
I'm reading a paper [R1] where the authors propose a MAP estimator for the phase noise and frequency offset. However equation (17), which I reproduce below, represents a challenging step for me and I ...
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666
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A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
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Ergodicity of a Markov chain
Hi,
I'd appreciate some help on a Markov chain result I'm trying to show. I believe the following is sufficient for a continuous time Markov chain $(X_t)$ with a countable state space to be ergodic:
...
- 181
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970
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Expected operator norm of inverse Wishart matrix
Let $ W\sim W_p(n,I)$ be a white $p\times p$ Wishart matrix, and assume $n>p+1$, which ensures that $W$ is invertible almost surely. Let $\|W^{-1}\|_{\text{op}}$ be the operator norm (maximum ...
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Random Unfoldings of the Cube
Motivated by unfoldings of the dodecahedron in How To Fold It --
How many (labeled or unlabeled) unfoldings of the 1 x 1 x n stack of unit cubes are there?
JORourke (4Nov16): John's original image is ...
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Proving that an optimal solution "converges"
This question is a follow-up on a previous question I asked at:
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 circle $C$ in the ...
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Spectra of VERY sparse random matrices
Consider an $n\times n$ random binary matrix $M$ with i.i.d. entries $m_{ij} \sim {\rm Bernoulli}(p)$, where $p = n^{-\beta}$ with $\beta \in (1,2)$. I am interested in the behavior of the singular ...
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Combinatorial Morse functions and random permutations
This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated space is a function ...
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Number of integer combinations $x_1 < \cdots < x_n$?
I asked this question earlier on math.stackexchange.com but didn't get an answer:
Let $0 < a_1 < \cdots < a_n$ be integers. Is there a closed formula (or some other result) for the number $N(...
- 16.2k
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A Variance-Tail Description for Continuous Probability Distributions
Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.
I would like to ask ...
- 24.7k
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symmetric difference of temperate zone and inscribed disk
For random domino tilings of the Aztec diamond of order $n$ or random lozenge tilings of the regular hexagon of order $n$, what's the typical order of magnitude of the area of the symmetric difference ...
- 19.7k
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2/3 power law in the plane
I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I ...
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A trick or a general technique? (Probabilistic Method)
Suppose we have some positive quantites $P$ and $Q$ which depend on some choices that we make, and we want to show that some choice makes the quotient $P/Q$ fall below some cool bound.
One idea is to ...
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797
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Fitting a mesh to a density function
Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
- 1,125
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shape of random q-weighted lattice path
Where can I find a detailed write-up of the asymptotic shape of a $q$-weighted Young diagram inside an $a$-by-$b$ box, especially one that uses a variational approach?
Equivalently, we can look at ...
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Is this probability distribution known in the literature?
In some work I was doing I derived a probability distribution that I do not recognize. Is it a known distribution?
$\Pr(X\le x)=\exp\left[-\frac{1}{2}\left(\frac{1}{2}x-\sqrt{1+\frac{x^{2}}{4}}\right)...
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Expectation of little o in probablity [closed]
If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \...
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Problem about expectation of maximum partial sum
Given a number $m$, a random composition (strong) of this number into $n$ positive parts so that we can get $n$ random variable $X_1, X_2,\dots, X_n$ with $$X_1+X_2+\cdots+X_n=m$$
Note that all ...
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Proof of Lomnicki and Ulam on infinite product probability spaces
Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set $F\...
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Many Brownian motions moving together
Let $ (B^i),\:{{i=1,\ldots,n}}$ be a set of independent Brownian motions. By $(X^i)$ we denote $(B^i)$ conditioned on the event
$|B^i_t-B_t^{i+1}|\leq 1,\quad \forall_{1\leq i\leq n-1}, \forall_{t\...
- 1,491
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Distribution of big component of set partitions
Consider the set $S_n = \{1, \dotsc, n\},$ and consider the set $P(n, k)$ of partitions of $S_n$ into $k$ parts (the cardinality of $P(n, k)$ is the Stirling number of the second kind $S(n, k).$ ...
- 96.4k
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Stochastic process describing long-term fluctuations
I need to model a process that has large, smooth and mean-reverting long-term fluctuations and some small short term wiggles, a sample path looks like this:
My first idea was to model it as an ...
- 667
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Interesting applications of [Martingale/Brown motion/diffusion/percolation ] theory?
This question is motivated by an exercise called "The Star-ship Enterprise's Problem" in Williams's book "probability with martingales", it can be stated as follows:
Suppose the control system on ...
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Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
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Time integral of an Ornstein-Uhlenbeck process
Let $X_t$ be an Ornstein-Uhlenbeck process solving $dx_t = \theta (\mu-x_t)\,dt + \sigma \,dW_t$.
The solution is known and given by:
$$ x_t = x_0 e^{-\theta t} + \mu(1-e^{-\theta t}) + \int_0^t \...
- 667
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Selecting two random points inside a sphere which are a fixed distance apart
Without appealing to a guess-and-check approach, how might I select a pair of random points inside of a sphere of radius $R$ s.t. the points always a distance $d \leq R$ apart? Can the selected ...
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Eigenvalue distributions of finite dimensional Wishart matrices
I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
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303
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Integrated colored Gaussian noise
Assume we have a colored Gaussian process $z_t$, with an autocorrelation function $cov(z_t,z_s)$ given by an analytical function $\alpha(t,s)$ (if it helps, one can assume that $\alpha(t,s) = \kappa e^...
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Characteristic polynomials of certain random symmetric matrices and the complexity of random Morse functions
Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with ...
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381
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Help prove a maximal inequality
Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number.
$S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$.
I want to prove:
$$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) \...
- 177
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466
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Maximum vertical distance for a lattice path when NSEW steps are allowed
Suppose we have a lattice path in 2D starting at the origin, in which north, south, east, and west steps are allowed. For a given path $L$, let $\max(L)$ be the maximum value of the $y$ coordinate ...
- 3,283
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198
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bounding the probability that a polynomial is near 0
Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
- 4,466
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729
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Density of countably additive measure in the set of all finitely additive measures.
Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...
- 6,034
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395
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computing average height-functions for lozenge tilings
Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? I prefer to ...
- 19.7k
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Uncertainty principle and Cramer-Rao bound - is there relation?
Just out of curiosity.
The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer ...
- 24.9k
7
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743
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Distribution of the sizes of conjugacy classes in the symmetric group.
This recent question makes me wonder: is there some known limit theorem for the distribution of the sizes of conjugacy classes in the symmetric group $S_n?$ A quick search seems to reveal nothing ...
- 96.4k