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.
9,023 questions
17
votes
4
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Sweep-segment bot: Will this random walk sweep the plane?
This model is inspired by the random behavior of the
Roomba sweeping robot.
Let a unit segment $ab$ in the plane be placed
initially with $a=(0,0)$ and $b=(1,0)$.
The segment is first rotated a ...
- 151k
1
vote
0
answers
265
views
"Lift and project" procedure for matrices
Definition. Let us call $n\times n$ matrix with non-negative entries good if sum of every row and column is equal to $1/n$.
Suppose we have a good matrix $A$. Let us consider the following strange "...
- 1,791
1
vote
1
answer
711
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Sequential sampling of Gaussian and von Mises-Fisher Random Variable
I don't find any article discussing this problem, so I dare to ask it.
Suppose we are dealing with a data $x_0 \in \mathbb{R}$ and a function $f:\mathbb{R} \to \mathbb{R}$. Say we repeatedly apply $f$...
7
votes
1
answer
5k
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Parametric vs Non-parametric Estimation of Quantiles
Motivation
Suppose that we need to estimate the median from a normal distribution with known variance. One non-parametric approach is to use the sample median as an estimator. However, this does not ...
- 197
9
votes
1
answer
1k
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A Game of Knights and Queens
Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such ...
- 26.9k
6
votes
1
answer
653
views
Change of space-time in Walsh's stochastic integral
One can read about Walsh's construction of martingale integral in the paper (pp.16-23)
http://www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf (Wayback Machine)
For $U,V\in \mathcal{B}(\mathbb{R}\...
7
votes
2
answers
4k
views
Commuting supremum and expectation
Given a one-parametric random function on a probability space $(\Omega,\mathcal F,\mathbb P)$:
$X:U\times\Omega\to \mathbb R \text{ and } (a,w)\mapsto X(a,w), \text{ with } \sigma(X(a))\subseteq \...
4
votes
1
answer
2k
views
Inequality on probability distributions
I would like to know if the following inequality is satisfied by all probability distributions (or at least some class of probability distributions) for all integer $n \geq 2$.
$\int_0^{\infty} F(z)^...
- 97
3
votes
0
answers
696
views
Expectation, multinomial distribution, and monotonicity (A conjecture)
Let $n$ and $k$ be two positive integers. Let $S = \{ \mathbf{p} \in \mathbb{R}^k : \mathbf{p} \geq 0, \sum_{i=1}^k p_i = 1 \}$ (i.e., a simplex).
Consider a function $\mathbf{f}:\mathbb{Z}^k \...
- 117
7
votes
0
answers
514
views
Why are low order Fourier coefficients more important for real-life probability?
Suppose have a probability distribution over space of $n$ binary variables. We can view logarithm of the density as a function $f$ from $\{-1,1\}^n$ to $\mathbb{R}$. Fourier expansion of the $f$ can ...
- 2,252
3
votes
1
answer
663
views
Stationary non-isotropic spatial stochastic processes
I asked this question in math.stackexchange but got no response;
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic?...
- 549
2
votes
0
answers
1k
views
Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution?
In Science and Hypothesis, chapter XI, The calculus of probabilities, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/...
0
votes
2
answers
722
views
Concentration bound using Azuma's inequality and Law of total probability
Given a function $f(X_1,\cdots,X_n,Y)$ on random variables $\{X_i\}$ and $Y$, which is continuous ,
I want to
show that $f$ concentrates around its expectation $\operatorname*{E}[f]$, i.e., a formula ...
10
votes
2
answers
1k
views
Random rotations in SO(3) and free group
Is it true that two random (w.r.t. Haar measure) rotations in $SO(3)$ generate a free group?
- 3,323
6
votes
2
answers
836
views
Probability of the maximum (Levy Stable) random variable in a list being greater than the sum of the rest?
Given a list of identical and independently distributed Levy Stable random variables, $(X_0, X_1, \dots, X_{n-1})$, what is the is the probability that the maximum exceeds the sum of the rest? i.e.:
...
- 513
18
votes
2
answers
4k
views
When is the function of a median closer to the median of the function than the mean of the function is to the function of the mean?
Background
notation: RV= random variable, $\mu=$ mean $m=$ median
Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.
If $f(...
- 225
5
votes
1
answer
781
views
Does a log-concave function on a convex set extend continuously to the boundary?
Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
- 8,512
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votes
5
answers
1k
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$E(X_1 | X_1 + X_2)$, where $X_i$ are (integrable) independent infinitely divisible rv's "of the same type"
The following is inspired by this recent question on math.stackexchange.
Two standard exercises in conditional expectation are to find ${\rm E}(X_1|X_1+X_2)$ where:
1) $X_i$, $i=1,2$, are independent $...
- 1,468
9
votes
1
answer
958
views
Quantitative bounds for multivariate central limit theorem
For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:
https://...
- 93
16
votes
4
answers
3k
views
"Uniform probability" on a set of naturals
It's an obvious and well-known fact that there is no uniform probability measure on a set of natural numbers (i.e. the one that gives the same probability to each singleton).
On a recent probability ...
- 349
16
votes
8
answers
4k
views
Brownian bridge interpreted as Brownian motion on the circle
Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle?
The Brownian bridge has some strange connections with the Riemann zeta function (see Williams'...
- 1,666
4
votes
1
answer
799
views
Results for Hitting Times of (Not Stationary) Ito Processes
Let $W_t$ denote the Wiener process and let
$$
dX_t = a(t, X_t) dt + b(t, X_t) dW_t
$$
be an one dimensional Ito SDE (stochastic differential equation, see Ito stochastic calculus).
A hitting time $...
- 1,544
10
votes
2
answers
2k
views
How to sample pairwise independent gaussians
If $X_1, \ldots , X_k$ are i.i.d normal random variables with mean $0$ and variance $1$, then is there a way to sample $Y_1, \ldots , Y_m$ for $m=\omega(k)$ such that each of the $Y_i$'s is a normal ...
- 563
3
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?
Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^*$ denote the space ...
- 8,512
7
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
- 8,512
0
votes
1
answer
801
views
Information criteria for ridge regression
Hi -- is there any analogue or adjustment of, say, Schwartz Bayesian (or other) information criterion that would be applicable to model selection in ridge regression with a given ridge parameter $\eta$...
- 177
16
votes
2
answers
4k
views
Is the space of continuous functions from a compact metric space into a Polish space Polish?
Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ ...
user6096
24
votes
6
answers
3k
views
Shortest grid-graph paths with random diagonal shortcuts
Suppose you have a network of edges connecting
each integer lattice point
in the 2D square grid $[0,n]^2$
to each of its (at most) four neighbors, {N,S,E,W}.
Within each of the $n^2$ unit cells of ...
- 151k
45
votes
1
answer
6k
views
Anti-concentration bound for permanents of Gaussian matrices?
In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
- 9,833
0
votes
0
answers
293
views
Open Jackson network with deterministic arrivals.
Dear Friends,
Is there any known Jackson-like theorem for an open Jackson network with deterministic arrivals?
Thanks,
Michael.
- 85
1
vote
1
answer
294
views
Stability of discrete queue (new twist)
Hi, I am new to queueing theory. I am interested in a question that I feel should be fairly basic, yet I haven’t really found a clear solution to it. Hopefully somebody here can help me.
We have a ...
- 501
3
votes
2
answers
833
views
Two geometric probability questions (one answered, one more to go)
Given $n$ independent uniformly distributed points on $S^2$, what's the distribution of the distance between two closest points?
Consider $n$ iid uniform points on $S^1$, $Y_1, \ldots, Y_n$, in ...
- 4,466
11
votes
2
answers
880
views
Covering a random graph with spanning trees.
Let $G=(V,E)$ be a connected graph, say $V=\{1,\ldots,n\}$. Let $F=(V,E')$ be a uniformly random forest in $G$. (In other words, $E'$ is a subset of edges $E$ not containing a cycle, and it is ...
- 4,922
14
votes
5
answers
4k
views
Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?
The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
- 943
1
vote
2
answers
237
views
Strongly correlated? Terminology question
Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega_1$, $\omega_2$, we have
$$
X(\omega_1)\le X(\omega_2)\quad\Longrightarrow\quad Y(\omega_1)\le Y(\...
- 24.8k
7
votes
1
answer
357
views
maximal coordinate on a sphere
What is the easiest (preferably without calculations) way to see that the mean value of $\max(x_1,x_2,\dots,x_n)$ on the sphere $\mathbb{S}^{d-1}= \{ (x_1,\dots,x_n):\ x_1^2+\dots+x_n^2=1 \}$ behaves ...
- 109k
0
votes
0
answers
319
views
Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
10
votes
2
answers
2k
views
Convergence of an empirical distribution w.r.t. the Hellinger distance
Let $P$ be a probability distribution on a finite set $\mathcal{X}$ and let $X_1, X_2, \ldots, X_n$ be drawn i.i.d. according to $P$. Define the empirical distribution:
$\hat{P_n}(x) = \frac{1}{n} \...
- 165
0
votes
1
answer
578
views
One-Variable Optimization Problem
$W_{opt}=\arg \{\max(\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha )\}$
subject to $\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$
We should find analytically the optimal $...
- 23
2
votes
1
answer
186
views
scalar diffusions are reversible
It is well known that under mild assumptions a scalar diffusion $dX_t = a(X_t) dt + \sigma(X_t) dW_t$ with invariant probability distribution $\pi$ is reversible. This is indeed not true for ...
- 2,133
9
votes
1
answer
1k
views
Can a non-Borel set be a standard Borel space?
Recall that a standard Borel space is a measurable space $(X,\mathcal{M})$ (i.e. a set with a $\sigma$-algebra) such that there exists a 1-1 bimeasurable map $\phi$ from $(X,\mathcal{M})$ to $[0,1]$ (...
- 29.8k
1
vote
1
answer
356
views
Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
- 95
10
votes
1
answer
1k
views
Bounds on $\|P^{k+1} - P^k\|$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k\gg n$
The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...
- 303
27
votes
7
answers
30k
views
When do 3D random walks return to their origin?
The probability of a random walk returning to its origin is 1 in two dimensions (2D) but only 34% in three dimensions: This is Pólya's theorem. I have learned that in 2D the condition of returning to ...
- 151k
1
vote
1
answer
722
views
Combination of probability distributions with maximum relative entropy
How can we figure out the combination of probability mass functions p and q, such that the relative entropy $D(p||q) = \sum p \log \frac{p}{q}$ is maximum? Of course this is a convex function.
I am ...
- 141
10
votes
2
answers
602
views
What is the probability that every pair of students is at some point in the same classroom?
A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 years, ...
- 226
2
votes
2
answers
391
views
linear ordering of color balls
Suppose that $n+m$ balls of which $n$ are red and $m$ are blue, are arranged in a linear order, we know there are $(n+m)!$ possible orderings. If all red balls are alike and all blue ball are alike, ...
- 123
3
votes
4
answers
3k
views
Probability Theory, Chernoff Bounds, Sum of Independent (but not identically distributed) r.v
Is there any known bound on sum of independent but not identically
distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in
the Chernoff bounds for ...
- 85
9
votes
2
answers
616
views
construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
- 1,839
0
votes
2
answers
294
views
Relationship between these two probability mass functions.
If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows:
$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $
what can ...
- 141