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.
9,026 questions
1
vote
1
answer
555
views
The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
- 55
0
votes
1
answer
262
views
Conditional Density of Random Variables
Hi all,
I read recently that for any three continuous random variables, X,Y and Z, the conditional densities are related by the following formula:
$p(x|y) = \int g(x| z) h(z | y ) dz $
where $p(x|...
- 37.7k
3
votes
3
answers
2k
views
How to bound the sup norm of a Rademacher process or equivalently a Gaussian process?
I want to know how to find an upper bound of the following expectation taken for both $t$ and $y$ as
$$\mathbb{E}\sup_{x \in D} \left|\sum_{k=1}^n t_k x^T y_k\right|,$$
where $D$ is the set of ...
- 13k
11
votes
2
answers
2k
views
Expected values of traces of products of random matrices
Suppose I want to compute a quantity of the type:
$\mathbb{E}\mathrm{tr}(AUBU^{\ast})$
where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher ...
- 1,890
6
votes
1
answer
566
views
Area of union of random circles in a plane
If $n$ circles of radius $r$ are drawn at random in the plane such that their centers lie in some region $S$ (we could take $S$ to be a square), what is the expected area $E$ of their union?
Edit: In ...
- 1,676
5
votes
0
answers
202
views
hitting time of a subset
Suppose you have the simple random walk on $L=\mathbb{Z}^n,$ and $A \subset L$ is a subset (in my case, the subset is a union of hyperplanes, so in particular infinite). There is a random variable $T(...
- 96.4k
1
vote
2
answers
191
views
Is there a notion of likelihood that incorporates information content?
Consider a random variable $F$ with a distribution parameterized by $\theta$ and another random variable $G$ with a distribution parameterized by a variate of $F$, denoted $f$. Note that $F$ is ...
- 2,600
0
votes
1
answer
182
views
How to Rigorize an inequalities argument
Context
I'm working on a problem involving Lovasz Local Lemma, for proving that there exists a graph with a certain property.
What I need to prove:
There exists some constant $c$, and functions $p,...
- 109k
8
votes
3
answers
1k
views
Expected distance between two points in the plane
Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$...
- 1,902
1
vote
1
answer
148
views
Staggered timing on 2-D random walks by multiple agents
In 2-D lattice random walks by multiple drunks who can't step onto each other, mathematically I would just say the whole cellular automaton updates "at once".
But to simulate this on a computer, I ...
CommunityBot
- 1
2
votes
1
answer
235
views
The distance between the centroid of $P$ points and the centroid of a subset of the points
Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points ...
- 96.4k
14
votes
2
answers
2k
views
Markov chains: invariant measures and explosion
The following seems like such an elementary question, but I didn't get anywhere with it.
Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure (...
- 3,069
0
votes
1
answer
407
views
Transformation of probability space.
Let (\Omega, F, P) be a probability space, which may have atoms (important), S be a set of measure-preserving transformations T:\Omega\to\Omega, that is, such that preimage T^{-1}(A) is measurable ...
- 1
3
votes
1
answer
599
views
Is positive part of the kernel measurable?
Let $(E,\mathscr E)$ be a measurable space and $Q:E\times \mathscr E\to\Bbb [-1,1]$ be a signed bounded kernel, i.e. $Q_x(\cdot)$ is a finite measure on $(E,\mathscr E)$ for any $x\in E$ and $x\mapsto ...
CommunityBot
- 1
1
vote
1
answer
132
views
Approximating Moment of Sum of RVs
Given
$X_i$ are independent random variables.
$|X_i| < 1$
$E[X_i] = 0$
$X = \sum_i^n X_i$
$var(X)=\sigma$
Prove:
$$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p
Things I've tried:
...
- 11.4k
14
votes
2
answers
783
views
Are two probability distributions uniquely constrained by the sum of their p-norms?
Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, ...
- 39.1k
3
votes
1
answer
5k
views
With Huffman code, why do we still need Shannon code?
I'm studying information theory by myself.
I'm confused about that since we already have Huffman code, which is the optimal code method, why are Shannon code and some other code still useful?
I ...
- 32.5k
1
vote
0
answers
223
views
Why this two model have same probability distribution?
(1)
Consider the following method of generating a random tree with $n$ nodes.
First expand the root node into two branches.
Then expand one of the two terminal nodes at random.
At time $k$, ...
- 351
3
votes
1
answer
505
views
Large deviations for sums of exponentially distributed random variables.
Take a large integer $R$, and let $(X_j)_{j\geq R}$ be a sequence of exponentially distributed random variables with parameters $\pi_j := j^{1+\alpha}$ ($\alpha>0$), so that $\sum_{j\geq R} \frac{1}...
- 96.4k
7
votes
0
answers
453
views
Does the law of a Feller Process on a non-locally-compact Polish space depend continuously on the initial condition (in Skorohod path-space)?
I am sure this is written down somewhere but cannot find it.
Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a ...
- 888
2
votes
1
answer
873
views
Bochner's Theorem and Total Positivity
Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with ...
CommunityBot
- 1
0
votes
1
answer
101
views
multimodal circular model
Hi, can someone provide me with a list of probability models that is akin to Von Mises but consists multiple (potentially infinite) modes that takes into account attractors in the entire 2-D spatial ...
CommunityBot
- 1
1
vote
0
answers
93
views
Potentials of class D
A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...
- 667
3
votes
0
answers
765
views
Bound on a sum involving binomial distribution
Let $f^B_{j,a}(s)$ be the probability mass function of the binomial distribution, that is $f^B_{j,a}(s) = {j \choose s} a^s (1-a)^{j-s}$. And let $F^B_{j+1,b}(s)$ be the cdf of the binomial ...
- 111
2
votes
2
answers
571
views
Family of Brownian Motions
I am trying to show the following statement
Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $x \in D$. Then $\Pi$ ...
- 279
3
votes
2
answers
430
views
Itô's Formula on a bounded Domain
Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost ...
- 279
3
votes
2
answers
352
views
Continuity of hitting distributions
Hi everybody
Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...
- 279
2
votes
1
answer
646
views
A wrong proof of Squared Bessel process
The squared Bessel process with $\delta$-dimension for $\delta>0$,
denoted by $BESQ^\delta(y)$, is given by
$$d Y_t = \delta t + 2 \sqrt{Y_t} d B_t, \ Y_0 = y\ge 0$$
where $B_t$ is BM under $(\...
- 5,005
7
votes
4
answers
1k
views
Recent impressive combinatorial developments in probability theory
In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv)
... I suspect that, for at least a decade, the most ...
- 1,302
2
votes
1
answer
447
views
MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
- 101
0
votes
2
answers
763
views
multivariate distributions unaffected by unitary transformations
Hi,
In my research I reached some very nice results for IID complex Gaussian vectors $\bf{x}$.
Now I realize that my results hold for any random vectors that are unaffected by a unitary map, i.e., $\...
- 5,005
4
votes
0
answers
445
views
Hessians of Fourier transforms of positive radial functions
$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\eW}{\mathscr{W}}$
While investigating the distribution of critical points of random funtions on tori I was lead to ...
- 34.7k
4
votes
2
answers
427
views
Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
CommunityBot
- 1
9
votes
2
answers
646
views
Rain droplets falling on a table
Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
- 28k
3
votes
1
answer
253
views
Bounds for duplicate finding with limited independence
(This is a follow up to this previous question on math.stackexchange.com.)
Assume a process that samples uniformly at random from the range $[1,\ldots,n]$. I am interested in the time to find a ...
CommunityBot
- 1
2
votes
0
answers
113
views
Does this series stopping times marching forward?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider stochastic differential equation
$$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$
Note that, the above SDE has a strong non-negative ...
- 1,399
7
votes
0
answers
438
views
Extracting particles from a determinantal point process
Consider $N$ real random particles $x_1,\cdots, x_N$ distributed according to a density $\rho(x_1,\ldots,x_N)$ with respect to the Lebesgue measure on $\mathbb R^N$, which is assumed to be invariant ...
- 2,135
7
votes
1
answer
579
views
Random Functions and Transition Probabilities
Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A transition probability from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability ...
CommunityBot
- 1
15
votes
2
answers
10k
views
Convergence of moments implies convergence to normal distribution
I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
- 5,005
3
votes
1
answer
376
views
The degrees in a random subgraph
Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
- 28k
8
votes
3
answers
431
views
Characterising semi-definite positiveness on vectors with non-negative entries
My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
- 52.3k
1
vote
0
answers
217
views
Calculating or estimating a combinatorial multivariate sum
Dear all,
I'm currently looking at a problem in which the following combinatorial product emerges:
$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\frac{m_1 !}{(m_1-n_1)!}\frac{(m_1+m_2-n_1)!}{(m_1+m_2-...
CommunityBot
- 1
8
votes
2
answers
1k
views
Does infinite-dimensional Brownian motion live in hyperplanes?
I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ ...
- 29.8k
0
votes
0
answers
352
views
prewhitening (whitening transform) in terms of expected-value-wr-sigma-algebra
I'm trying to understand the mathematics of prewhitening a little better. (See http://en.wikipedia.org/wiki/Whitening_transformation, e.g.)
Taking the conditional expectation of an RV with respect to ...
17
votes
1
answer
787
views
Homotopy of random simplicial complexes
A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with ...
- 13.6k
2
votes
1
answer
272
views
Derivative of the CDF of a family of random variables
Suppose I have a r.v. $Z = X + \alpha Y$ and that $F_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = ...
- 54.2k
0
votes
0
answers
161
views
T. Lyons Criterion
Hello all,
I want to prove that any flow on the following tree must have an infinite energy.
The structure of the graph is (taken from R.Lyons and Y.Peres book)
"We’ll construct a tree $T$ embedded ...
- 1
2
votes
0
answers
124
views
Does a certain Theorem on Boltzmann Distributions exist?
Suppose $X_n(z)$ is a sequence of random variables with a boltzmann distribution on $\{1,2,\dots n\}.$ That is $$P(X_n(z)=j)=\frac{c_{j,n} z^j}{F_n(z)}$$ where $F_n(z)=\sum_{j=1}^n c_{j,n}z^j$ is a ...
- 1,306
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)^...
- 5,721
1
vote
1
answer
390
views
Probability that p and q are both prime provided q-p=2r
Hello,
I would like to know whether there is a way, thanks to the prime number theorem, to give some kind of an equivalent of the probability that two positive integers $p$ and $q$ less than a given ...
- 30.1k