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.

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A variant of random walk

Standard random walk assumes a sequence of iid RVs $\{X_i\}_{i\geq 0}$ and studied the distribution of $S_n=\sum_{i=0}^n X_i$. Here, I am wondering whether there is some work on $T_n=\sum_{i=0}^n \...
maomao's user avatar
  • 492
16 votes
0 answers
819 views

Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks on $\mathbb{Z}^2$ that turn left or right at each step (i.e., they cannot continue the direction of the preceding step) have been ...
Joseph O'Rourke's user avatar
1 vote
2 answers
504 views

Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that the number of twin primes below $x$ should be roughly $\dfrac{x}{\...
Sylvain JULIEN's user avatar
9 votes
1 answer
4k views

Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$. Fix $N$ and consider now a discrete version ...
Oleg's user avatar
  • 911
25 votes
4 answers
10k views

Rate of convergence in the Law of Large Numbers

I'm working on a problem where I need information on the size of $E_n=|S_n-n\mu|$, where $S_n=X_1+\ldots+X_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX_1$. For this to make sense, the $(...
Anthony Quas's user avatar
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53 votes
4 answers
3k views

When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...
Eric Naslund's user avatar
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2 votes
1 answer
738 views

Push-forward of sum of two maps

Let $X=R^n$ and $Y=R^m$ be two Euclidean spaces with $m<n$. Let $\varphi$ and $\phi$ are two smooth maps from $X$ to $Y$, and $\mu$ a probability measure on $X$. Is there any relationship between $(...
Hongdmhut Hong's user avatar
4 votes
1 answer
434 views

Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
Nate Eldredge's user avatar
7 votes
1 answer
179 views

Is there a Degenerate Dependency Local Lemma?

The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded. Here I ask whether another possible generalization (for which I could not yet ...
domotorp's user avatar
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9 votes
1 answer
371 views

A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as $$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...
math-Student's user avatar
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9 votes
1 answer
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Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
Chee's user avatar
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2 votes
0 answers
86 views

Bounds on number of distinct substrings

I have a table with $r$ rows of length $\ell$, with each cell containing a letter from an alphabet $A$ of length $a$. I'm trying to determine the expected number of distinct strings of length $k$ ...
akuan30000's user avatar
5 votes
3 answers
702 views

Is there an analytic solution for this partial differential equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial t}=-\frac{\partial}{\partial\theta}\Big[[\sin(k\theta)+f]P(\theta,t)-D\frac{\...
Jay's user avatar
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4 votes
4 answers
1k views

Higher Moments, what are they good for? [closed]

Absolutely nothing? And now seriously - When I studied the basics of probability theory, and even in more advanced topics (random walks, stochastic processes, etc.), I always felt that the mean and ...
Miel Sharf's user avatar
2 votes
1 answer
244 views

Hitting probabilities for conditioned oriented random walk monotonic?

Consider an oriented random walk on $\mathbb Z^2$ (i.e. only steps $\rightarrow$ and $\uparrow$ with equal probability.) Say we let the walk go $2m$ steps then start guessing sites at distance $2m$ ...
mathjunge's user avatar
  • 181
2 votes
1 answer
292 views

Criterion for weak convergence of probability measures on S' or D'

Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...
MathNovice's user avatar
2 votes
0 answers
164 views

Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs. It is known that X and Y have the same CDF iff they have the same MGF. My ...
axk's user avatar
  • 517
2 votes
2 answers
153 views

Do all positive distributions on $N$ variables factor pairwise?

The Hammersley-Clifford theorem says that any positive probability distribution satisfies one of the Markov properties with respect to an undirected graph G if and only if its density can be ...
wrvb's user avatar
  • 123
10 votes
1 answer
509 views

a question on 0-1 valued stochastic process

Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm ...
Arkadi Predtetchinski's user avatar
6 votes
0 answers
199 views

Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a ...
Bjørn Kjos-Hanssen's user avatar
13 votes
0 answers
411 views

Transitivity of balanced mass transport in Z

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
James Propp's user avatar
  • 19.4k
2 votes
1 answer
135 views

Variant of Skorokhod's theorem

Consider the following situation: $S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful). There is a a random variable $\zeta: \Omega \to S$. $f_n(\zeta) \to^d \eta$, i....
Nate Ackerman's user avatar
2 votes
0 answers
248 views

Smallest Singular Value of a Random Matrix with Dependent Entries

Overview I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
QAMS's user avatar
  • 98
2 votes
1 answer
444 views

Weak convergence of probability measures on weak versus strong dual

The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
Abdelmalek Abdesselam's user avatar
7 votes
0 answers
496 views

Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance $$ \mathbb{E} [\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)} \widehat{g}(\xi)}{|\xi|^{d-2[\...
Abdelmalek Abdesselam's user avatar
6 votes
2 answers
477 views

Tail sigma-algebra of a branching random walk

I am looking for any known results about the tail sigma-algebra of a branching random walk. To be specific, let $T$ be the nodes of an infinite binary tree rooted at $r \in T$. Let $\{X_t\})_{t \in T}$...
Vladimir's user avatar
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4 votes
1 answer
1k views

Estimating the distribution of minimal hamming distances within a set of strings?

Is their an efficient mathematical way to estimate the distribution of minimal hamming distances for a set of random strings of length 8 over a 4-letter alphabet? E.g. given a set of 100-10,000 ...
queryous's user avatar
14 votes
2 answers
940 views

The power of two random choices with pairwise independence

Throw $n$ balls into $n$ bins, and let $X_n$ be the max load. That is the number of balls in the fullest bin. It is known that if the balls are thrown uniformly and independently at random then $\...
Simd's user avatar
  • 3,195
1 vote
1 answer
147 views

Probability Content of a random ball in R^n

As a follow up to this question, concerning this paper: Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...
Skrodde's user avatar
  • 329
8 votes
1 answer
326 views

Transitive closure of balanced mass transport in Z (move to close)

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
James Propp's user avatar
  • 19.4k
14 votes
1 answer
1k views

Natural probability on integers

This is a follow-up to this classical question asked recently here: we know (e.g. using the second Borel-Cantelli Lemma) that no probability measure on $\mathbb{Z}$ has the property that $n\mathbb{Z}$ ...
Benoît Kloeckner's user avatar
3 votes
2 answers
714 views

Moment problem for discrete distributions

Let $x_1, \dots, x_N \in \mathbb R$ and consider the discrete distribution $\mu := \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$, where $\delta_x$ denotes the Dirac measure, i.e. for any measurable set $B \...
user avatar
18 votes
3 answers
1k views

Existence of a "quasi-uniform" probability distribution on $\mathbb{Z}$

Does there exist a probability distribution on $\mathbb{Z}$ such that for every integer $n\geq 1$, the probability that a random integer $x$ is divisible by $n$ equals $1/n$? Henry Cohn has an ...
Richard Stanley's user avatar
-2 votes
1 answer
295 views

If $(X_n+Y_n)$ has bounded variance, is the same true for $(X_n)$ and $(Y_n)$? [closed]

Let $(X_n)$ and $(Y_n)$ be two sequences of random variables defined on the same probability space such that the variance of all components $X_n$, $Y_n$ is finite and the sequence of variances of $X_n+...
herrsimon's user avatar
  • 235
3 votes
1 answer
599 views

Projective family of probability spaces

This is a crosspost of this question from MSE. I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions $f_{tt}=1_{...
Exterior's user avatar
  • 905
2 votes
0 answers
1k views

Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)...
Bullmoose's user avatar
  • 897
7 votes
1 answer
1k views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let $\pi(\...
Herr K.'s user avatar
  • 173
5 votes
2 answers
255 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
geodude's user avatar
  • 2,129
7 votes
0 answers
621 views

Concentration of measure for uniform distribution on Stiefel manifolds

This is my first post on MO, so I hope the question is suitable. I am looking at the uniform distribution on the Stiefel manifold, but more specifically, at the uniform distribution on the ...
Math-user's user avatar
  • 203
5 votes
1 answer
218 views

Average probability that a random cosine polynomial with bernoulli coefficients is small

Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any $\...
TOM's user avatar
  • 2,218
4 votes
2 answers
197 views

Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles. It sounds intuitive to say that deviations from the mean ...
Cartucho's user avatar
4 votes
0 answers
164 views

Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...
Cristián Antuña's user avatar
1 vote
1 answer
125 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$...
Pablo's user avatar
  • 11.2k
6 votes
1 answer
257 views

Sufficient conditions for establishing a total order on a family of probability distributions?

Let $\mathcal{X}$ be some set of independent random variables. Define the ordering on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there ...
Steven Pav's user avatar
1 vote
1 answer
207 views

Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s? So far I only figured out that I can do Monte ...
nivwusquorum's user avatar
3 votes
2 answers
273 views

Expectation of Gaussian random vector & arbitrary function thereof?

I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity: where the <.> operator refers to a population average. No source or ...
DankMasterDan's user avatar
3 votes
1 answer
1k views

Proof for additivity of cumulants

If one does not define cumulants via the cumulant generating function (cgf), e.g. because the cgf does not exist, then an alternative way is to use the recusion \begin{align*} \kappa_n=\mu'_n-\sum_{m=...
user avatar
1 vote
1 answer
136 views

Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
Michael's user avatar
  • 263
2 votes
1 answer
170 views

How to choose a random proper coloring

I am studying proper colorings of complete bipartite graphs and I'd like to be able to pick a random proper coloring and the compute some things about it. Recall that a proper coloring of a complete ...
user43928's user avatar
  • 175
5 votes
0 answers
119 views

Real Zeros - tail estimate

Given a random polynomial with Gaussian coefficients, the Kac-Rice formula tells us what the expected number of real zeros is (for more on this, see the excellent paper of Edelman and Kostlan in the ...
Igor Rivin's user avatar
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