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.
9,027 questions
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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 ...
7
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579
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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 ...
7
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122
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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 ...
7
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171
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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 ...
7
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763
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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 ...
7
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774
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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 ...
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209
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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 ...
7
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550
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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) ...
7
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3k
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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 ...
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759
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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 ...
7
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395
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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
$$
f(x)=\frac{1}{|B(x,r)...
7
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179
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Can one "smooth over" k-wise independence to get actual independence?
I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...
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267
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Can primes be (almost) random sequence in von Mises sense?
Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
7
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497
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Extreme unitary minimal models of conformal field theory
Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...
7
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245
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Distribution of trivial subset sums
Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
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519
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Squaring random Schwartz distributions
Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
\widehat{g}(\xi)}{|\xi|^{d-2[\...
7
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678
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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 ...
7
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1k
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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 ...
7
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141
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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 $\epsilon>0$. ...
7
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627
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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 ...
7
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129
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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?
7
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280
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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 ...
7
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639
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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^...
7
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620
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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 ...
7
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300
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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 ...
7
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454
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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 ...
7
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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^...
7
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438
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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 ...
7
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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; ...
7
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0
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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 ...
7
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0
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396
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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 ...
7
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2
answers
1k
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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 \...
7
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0
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514
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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 ...
7
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718
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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_{...
7
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0
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3k
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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.
7
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1
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391
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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(...
6
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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 ...
6
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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(...
6
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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 ...
6
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2
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2k
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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,...
6
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4
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2k
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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 ...
6
votes
2
answers
2k
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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 "...
6
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2
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2k
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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 ...
6
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4
answers
452
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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 (...
6
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2
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2k
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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}[|...
6
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2
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608
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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?
6
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2
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775
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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, ...
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 ...
6
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3
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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?
6
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4
answers
1k
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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 ...