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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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Probability of landing inside the convex hull of previously sampled points

Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$. I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...
Maxim's user avatar
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Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
Tom Copeland's user avatar
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Discrepancy of the finite approximation of the Lebesgue measure

Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
Fedor Petrov's user avatar
7 votes
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171 views

What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?

Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
Elle Najt's user avatar
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7 votes
1 answer
763 views

Reference request: discretisation of probability measures on $\mathbb R^d$

Given a probability measures $\mu$ on $\mathbb R^d$ with finite first movement, i.e. $$\int_{\mathbb R^d}|x|\mu(dx)~~<~~+\infty.$$ My concern is to approximate $\mu$ some $\mu_n$ that is ...
user111097's user avatar
7 votes
0 answers
774 views

Calculate the expectation of the maximum of averaged random walks

Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$ Is ...
John Wong's user avatar
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209 views

Stabbing disks in space, or: Galactic alignment

I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are random within a sphere of radius $R>1$, and which are each oriented randomly. I'd like to find a line $L$ that ...
Joseph O'Rourke's user avatar
7 votes
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550 views

Counter-example to the completeness of the Wasserstein metric

$\newcommand{\P}{\mathcal{P}}$ Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
Oleg's user avatar
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3k views

What is vague convergence and what does it accomplish?

For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
Greg Zitelli's user avatar
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759 views

Product of two random Gaussian matrices - orthant probability

Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
Daniel Soudry's user avatar
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395 views

Fixed radius mean value property implies harmonicity?

Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent: $f$ is harmonic. $f$ satisfies the ball mean value property $$ f(x)=\frac{1}{|B(x,r)...
Snoop Catt's user avatar
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179 views

Can one "smooth over" k-wise independence to get actual independence?

I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...
untitled459's user avatar
7 votes
0 answers
267 views

Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
Bogdan Grechuk's user avatar
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497 views

Extreme unitary minimal models of conformal field theory

Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge $$ c=1-\frac{6}{m(m+1)}\ . $$ I ...
Abdelmalek Abdesselam's user avatar
7 votes
0 answers
245 views

Distribution of trivial subset sums

Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
Igor Rivin's user avatar
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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
7 votes
0 answers
678 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
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1k views

Do isonormal Gaussian processes have measurable sample paths?

Let $H$ be a real separable Hilbert space. Let $W=\{W(h):h\in H\}$ be a real-valued stochastic process defined on a complete probability space $(\Omega,\mathcal{F},P)$. Assume that $W$ is a centered ...
Jason Swanson's user avatar
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0 answers
141 views

Neigbourhood of Poisson point process

Let $C$ be a unit circle and $P$ be a Poisson point process on $C$ with intensity $1$. For each point $p$ of the process we draw an interval $(p-\epsilon, p+\epsilon)$ for some $\epsilon>0$. ...
Piotr Miłoś's user avatar
7 votes
0 answers
627 views

Elementary proof of lack of phase transition in Ising models with external fields

I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the ...
Anthony Quas's user avatar
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7 votes
0 answers
129 views

Speed on recurrent graphs

Suppose that $G$ is a (locally finite) graph such that the simple random walk on $G$ is recurrent. Does this imply any upper bound on the speed $\mathbb{E}d(X_t,X_0)$ of such random walk?
Marcin Kotowski's user avatar
7 votes
0 answers
280 views

Expected minimum Hamming distance with overlaps

Let's say we sample two random binary vectors, one called $A$ of length $n$ and the second called $B$ of infinite length. Now we compute $X_k= \min_{i\in[k]} w(A \oplus B[i,i+n-1])$ where $w$ computes ...
user avatar
7 votes
0 answers
639 views

When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R^...
user32372's user avatar
  • 241
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0 answers
620 views

Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
user avatar
7 votes
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300 views

Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
Tom LaGatta's user avatar
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454 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 ...
Wolfgang Loehr's user avatar
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0 answers
813 views

Wasserstein distance between two diffusion processes.

I would like to know if there exists a formula to compute the $L^2$-Wasserstein distance between the laws $P_1$ and $P_2$ in path space of two diffusion processes: $$dx_t=f_1(x_t)dt + \sigma_1(x_t)dB^...
user16215's user avatar
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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 ...
Adrien Hardy's user avatar
  • 2,135
7 votes
0 answers
440 views

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; ...
Tom Alberts's user avatar
7 votes
0 answers
743 views

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 ...
Igor Rivin's user avatar
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7 votes
0 answers
396 views

Stable distributions for Lindeberg exchange strategy?

Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
András Salamon's user avatar
7 votes
2 answers
1k views

Weighted Poincaré inequality

Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...
Alekk's user avatar
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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 ...
Yaroslav Bulatov's user avatar
7 votes
0 answers
718 views

Is there a continuous-time version of Kingman's subadditive decomposition theorem?

Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying: $x_{l,n} \le x_{...
Pablo Lessa's user avatar
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7 votes
0 answers
3k views

Good textbooks on probability and/or stochastic processes, emphasizing simulation

Any recommendations for textbooks on probability and/or stochastic processes that emphasize simulation? I'll be teaching this course in the Fall.
James Propp's user avatar
7 votes
1 answer
391 views

Idempotent splitting for Markov kernels

Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation, $$e(A|x) = \int_X e(...
Tobias Fritz's user avatar
  • 6,406
6 votes
3 answers
855 views

Series involving power of the index

How to prove the following identity $$ \sum_{n=1}^{\infty} \frac{n^{n-1} e^{-n}}{n!} = 1$$ analytically (which can be confirmed with $Mathematica$)? The standard trick for geometrical series does not ...
Jerry's user avatar
  • 77
6 votes
4 answers
1k views

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(...
Ralph's user avatar
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6 votes
10 answers
8k views

Best introduction to probability spaces, convergence, spectral analysis

I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP: probability ...
jefflovejapan's user avatar
6 votes
2 answers
2k views

Is there a name for this theorem?

I wonder if there is a name or reference for the following fact. It is not the proof I am looking for. Let $s_1, s_2, ...,s_n$ be non-negative real numbers ordered in a non-increasing way. Let $b_1,...
Miklos Bona's user avatar
6 votes
4 answers
2k views

Probability that randomly chosen integers from a restricted set of natural numbers are coprime

We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is $$ P(k) = \frac{1}{\zeta(k)}. $$ I am looking at a special case of ...
Nilotpal Kanti Sinha's user avatar
6 votes
2 answers
2k views

Expected maximum number of "prank cigarettes" in an average pack

"Real-life" motivation. The German satirical magazine Der Postillon suggested a few measures for deterring smokers from their bad habit. I especially liked the idea of inserting one "...
Dominic van der Zypen's user avatar
6 votes
2 answers
2k views

Are Gaussian Processes more important than other stochastic processes?

I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
s5s's user avatar
  • 87
6 votes
4 answers
452 views

Counting card distributions when cards are duplicated

If we have a deck of $48$ different cards and $4$ players each get $12$ cards, it is well known how to calculate the number of possible distributions: $\frac{48!}{12!12!12!12!}$ In a german card came (...
Horst Fickenscher's user avatar
6 votes
2 answers
2k views

Is there a universal bound for this ratio of expectations?

Let $X$ and $Y$ be two zero-mean independent and identically distributed random variables. Is there a bound for the following ratio, $$\frac{\mathbb{E}[|X+Y|]}{\mathbb{E}[|X|+|Y|]}=\frac{\mathbb{E}[|...
Math_Y's user avatar
  • 287
6 votes
2 answers
608 views

Whence “uniform distribution”?

The “Earliest Uses” site suggests that the expression “uniform distribution” first appeared in Uspensky (1937), and “uniformly distributed” in Sakamoto (1943). Is that true?
Francois Ziegler's user avatar
6 votes
2 answers
775 views

Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
user avatar
6 votes
2 answers
725 views

Threshold function for a graph not being planar

A graph property $\mathcal{P}$ is monotone increasing if $G\in \mathcal{P}$ implies $G+e \in \mathcal{P}$, i.e., adding an edge to a graph does not destroy the property. It is well-known that every ...
W. Paul Liu's user avatar
6 votes
3 answers
999 views

Does there exist an almost surely differentiable martingale?

Does there exist a continuous time martingale $X_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?
Nate River's user avatar
  • 6,323
6 votes
4 answers
1k views

Improvement of Chernoff bound in Binomial case

We know from Chernoff bound $P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where $X$ follows Binomial($N, \frac{1}{2}$). If I take $N=1000, \epsilon=0.01$, the upper bound is ...
user15864's user avatar
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