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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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 $2m$ ...
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1
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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 $< f,X>$ ...
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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 ...
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2
answers
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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 ...
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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 ...
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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 ...
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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 ...
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1
answer
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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....
- 2,209
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answers
249
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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 ...
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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 $S'(\mathbb{R}^d)$, ...
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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[\...
- 21.8k
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2
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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}$...
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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 ...
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2
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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 $\...
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1
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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. ...
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1
answer
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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 ...
- 19.4k
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1
answer
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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}$ ...
- 14.2k
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votes
2
answers
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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 \...
user45183
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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 ...
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votes
1
answer
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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+...
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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_{...
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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)...
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votes
1
answer
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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(\...
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votes
2
answers
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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 ...
- 2,129
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votes
0
answers
625
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 ...
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votes
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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 $\...
- 2,248
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votes
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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 ...
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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 ...
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vote
1
answer
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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.$$...
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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 ...
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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 ...
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answers
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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 ...
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answer
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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=...
user45183
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vote
1
answer
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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 ...
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votes
1
answer
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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 ...
- 175
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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 ...
- 95.6k
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votes
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Maximum occupancy balls in bins with limited independence
Throw $n$ balls into $n$ bins and let $X_n$ be the maximum occupancy. That is the maximum number of balls found in any bin.
If you throw the balls uniformly and independently it is known that $\...
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votes
2
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Anti-concentration for sums of geometric random variables
Consider the random variable $Y = Y_1 + \dots + Y_k$, where each $Y_i$ is iid distributed as a geometric random variable with sucess probability $p$; here we should think of $p$ as being close to zero....
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Variance of truncated normal distribution
Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. $...
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views
Existence of strong solution to SDEs with non-Lipschitzian drift
Consider the SDE:
$$dX_t=b(X_t)dt+dW_t\quad X_0=x$$
If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution.
I want to know if we assume $b$ ...
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votes
1
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Determining the Fourier transform
Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...
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votes
0
answers
142
views
Bounds on moving average process
Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables and define the average process $\{Y_t\}$ as
$$
Y_t = \sum_{i=1}^p a_k X_{t-i}
$$
with some constants $a_1,\cdots,a_p \in \mathbb{R}$. This ...
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votes
1
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views
Relatively compact sets in Ky Fan metric space
Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E \...
- 505
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0
answers
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Estimating the number of colors in a bucket
This question was previously posted to Math Stack Exchange here.
Suppose we have a bucket containing a large (but known) number of balls. Each of the balls has a color. We don't know how many colors ...
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votes
0
answers
107
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Is Wiener's Tauberian theorem true in Wiener space?
Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.
Is the following true?
...
- 3,970
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vote
0
answers
368
views
Topological properties of space of Radon measures
Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...
- 1,741
2
votes
2
answers
341
views
Is this a sufficient condition for joint normal distribution?
Suppose I have a random vector $\boldsymbol{Z}$, if I can prove that for $\forall \boldsymbol{\lambda} \neq \boldsymbol{0}$ where $\boldsymbol{\lambda}$ is a fixed vector, not a random vector,
$\...
- 399
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votes
1
answer
436
views
Smallest $k$ so that $k$-wise independence guarantees a constant expected minimum
Imagine you sample $n$ numbers with replacement uniformly from the integers $1,\dots, n$ (we can assume $n$ is large). Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but ...
- 3,195
3
votes
1
answer
618
views
Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$
Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...
- 31
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votes
1
answer
323
views
Berry-Esseen bound in 2 dimensions for linear combinations
Let us say have a sequence of $n$ 2-$D$ random variables $X_i=(\varepsilon_i/\sqrt{n},i\varepsilon_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon_{i}$ are independent random variables such that $\mathbb{P}(...
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