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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Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding "absolute separability" probabilities

Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$. Integration over $...
Paul B. Slater's user avatar
2 votes
1 answer
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Unit probability subset of image of a measurable set

Let $(\Omega,\Sigma,P)$ be a probability space and $A\in \Sigma$ be such that $P(A) = 1$. Let $X:\Omega \to \mathbb{R}^n$ be a $\Sigma$-measurable map and $\Lambda_X (B) = P(X \in B)~\forall B \in \...
jpv's user avatar
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Sum of independent random variables having exponential tails

Suppose $X_1, X_2,\dots, X_n$ are iid random variable,$P(X_i=-\infty)$ is allowed,$P(X_i>v)< e^{-v}\forall v>0$, $X$ is distributed as $X_i$, if $c$ is a finite real number such that $E(X)&...
Myshkin's user avatar
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Eigenvalue density for correlated samples

Suppose we generate $P$ samples $\mathbf{x}^1, \dots, \mathbf{x}^P$ from a multivariate normal distribution of dimension $N$, mean zero and covariance $\Sigma$. Let $\mathbf{C}=\frac{1}{P}\sum_{k=1}^...
valle's user avatar
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Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$. Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix. Define $S^k=...
Thomas Dybdahl Ahle's user avatar
4 votes
2 answers
145 views

Understanding equiprobable trinomial identity

With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
maliesen's user avatar
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Probabilistic lower bound on largest singular value of matrices

I have a distribution $\mathcal{D}$ that spits out vectors in $\{-1, 1\}^N$. Suppose I have a sample of $H$ of these vectors which I arrange into a matrix $M$ of the form $H \times N$. Consider the ...
Halbort's user avatar
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Coupling argument involved in the contracting and mixing properties of the Glauber dynamics for an Ising model

While doing a research work, I had to read about the Glauber dynamics for an Ising model. A wonderful account on this is given in the book Markov Chains and Mixing Times by Levin, Peres and Wilmer. ...
abcd's user avatar
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Exponential upper bounds for sums of martingale differences

Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}...
Oleksandr Z.'s user avatar
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168 views

Simple question regarding negatively associated random variables

I'm trying to learn about negative association of random variables. A definition can be found here: http://www.cs.cmu.edu/~dwajc/notes/Negative%20Association.pdf. Now, consider the following ...
christian.giovannie's user avatar
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475 views

Which random walk can generate gamma distribution in the limit?

Symmetric random walk, its probability distribution is binomial coefficient, in the continuous limit, is Gaussian distribution: $\displaystyle e^{- x^{2}}$ What kind of random walk, its probability ...
david's user avatar
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Strange inequality relating Binomial pmf and cdf

I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf. Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
user113925's user avatar
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How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$?

Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$: $$ \delta = \sum_{s=T}^{n} p^s (1-...
Ted's user avatar
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Expected minimum number

$X_i$ iid with $P(X_i=j)=p_j$, $j=1, \dots, m$. $\sum_{j=1}^m p_j = 1$. Define $N = \min\{n>0:X_n = X_0\}$, compute $E(N)$. I have two solutions, but different answers: Solution 1 $E(N) = E(N\...
gnohz's user avatar
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276 views

Convexity of the expectation of boolean functions

Let $$f:\{-1,1\}^n \to \{-1,1\}$$ be a monotone, odd ($f(-x)=-f(x)$) Boolean function. Let $$F:[0,1]\to[0,1]$$ denote the probability that $f(x_1,...,x_n)$ where $x_1,...,x_n$ are i.i.d. $\pm1$ R.V. ...
gidi's user avatar
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0 answers
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About martingales induced by iterative processes

Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \...
gradstudent's user avatar
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4 votes
2 answers
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Minimal coupling

Let $\mu,\nu$ be probability measures defined on a common measure space $(\Omega,\mathcal F)$. A coupling of $\mu,\nu$ is a probability measure $\pi$ on $(\Omega^2,\mathcal F^2)$ with marginals $\mu,\...
pre-kidney's user avatar
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Relationship between a certain binary optimal transport and total-variation of modified distributions

Let $\mathcal X$ be a Polish space, and let $(N_x)_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N_x \cap N_{x'} = \emptyset\...
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An extremal value distribution from monotone sequences

Pick two integer sequences $d>a_n\geq a_{n-1}\geq\dots a_1\geq0$ and $d>b_n\geq b_{n-1}\geq\dots b_1\geq0$ where $d$ is an integer bound with following method: Pick $a_n$ uniformly from $[0,d]$ ...
VS.'s user avatar
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4 votes
1 answer
317 views

Degenerate Gaussian Integral

I have an integral over a subspace of $\mathbb{R}^n \times \mathbb{R}^n$ with an integrand of the form $$\exp\left(-\frac{1}{2}\left[||u^2|| + \langle u, v \rangle + ||v||^2\right]\right)$$ The ...
DJA's user avatar
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1 answer
66 views

Comparing noisy truncated RV with noisy regular RV

For some reason, I'm having difficulties proving something that is intuitively simple. Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...
MRm's user avatar
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355 views

What is the status of the Born Rule in axiomatic QM?

While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
More Anonymous's user avatar
2 votes
1 answer
79 views

Series of random variables tending to infinity P.a.s. and in L^2

Let $(X_n)n \in \mathbb{N}$ be a sequence of random variables in $L^2(\Omega)$ verifying for any $n$ and for some positive real $K$ : $$ X_n \geq 0 \; \mathbb{P}-a.s., $$ $$ \mathbb{E}[Xn] \geq K >...
ALav's user avatar
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What is the measures monad for FDHilb?

I am labouring under a particular assumption that, perhaps, needs to be corrected. I believe that FDHilb, the category of Finite Dimensional Hilbert spaces and general linear maps is a category of ...
Ben Sprott's user avatar
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1 answer
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Size of the orbit of a dense set

This question is a follow-up to: this post. Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
MrsHaar's user avatar
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1 vote
1 answer
740 views

Ito's Formula for functions that are $C^2$ almost everywhere

In the conventional Ito's formula, it is required that $F$ is $C^2$ everywhere. However I've seen mentioning of a slightly weaker condition, where $F$ is $C^1$ everywhere but $C^2$ almost everywhere. ...
Jackie Lu's user avatar
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1 answer
114 views

What is the probability of an empty convex $k$-gon among many given points?

Given a finite number of points in the plane in general position, call a convex subset empty if its hull doesn't contain any other of the points. For a big number $n$ of randomly distributed ...
Wolfgang's user avatar
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5 votes
1 answer
426 views

Trying to understand Fisher's proof

$\newcommand{\al}{\alpha}$ For $i=1,\dots,n$, let \begin{equation} R_i:=\frac{X_i}{X_1+\dots+X_n}, \end{equation} where the $X_i$'s are iid standard exponential random variables. Let $$R_*:=\max_{1\...
Iosif Pinelis's user avatar
3 votes
1 answer
282 views

Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$

Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be ...
Ribhu's user avatar
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0 answers
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Empirical estimation of $\inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega)$, given i.i.d samples from $\mu$ and $\nu$

Let $\mathcal X$ be a Polish space and $\Omega \subseteq \mathcal X^2$ be open. Let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by $$ c_\Omega(\mu,\...
dohmatob's user avatar
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2 votes
1 answer
568 views

Sufficient condition for function of conditional probability density to be increasing

Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
Ararat's user avatar
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1 vote
1 answer
196 views

upper bound of the expectation of upper quantile ratio

We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$, where $\mathcal{D}$ is a distribution ...
Amit Portnoy's user avatar
12 votes
2 answers
945 views

How rare are unholey permutations?

For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$. Given a permutation $\pi$ of $[n]$, we define the holeyness $D(\pi)$ of $\pi$ as being $...
H A Helfgott's user avatar
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3 votes
0 answers
147 views

Beta distribution and Wiener process

Suppose $W(t)$ is the standard Wiener process on $[0; 1]$ and $\{T_x\}_{x \in \mathbb{R}}$ is a collection of random variables defined by the following relation: $$T_x = \mu(\{t \in [0;1] | W(t) > ...
Chain Markov's user avatar
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2 votes
0 answers
318 views

Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?

In Bobkov and Tetali - Modified Log-Sobolev Inequalities, Mixing and Hypercontractivity (extended version Modified Logarithmic Sobolev Inequalities in Discrete Settings), at the beginning of section 3,...
Ella Sharakanski's user avatar
4 votes
0 answers
210 views

How frequent are permutations with small interleaving?

For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$. Let $\pi$ be a permutation of $[n]$. For simplicity, assume that $\pi$ is an $n$-cycle. ...
H A Helfgott's user avatar
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5 votes
2 answers
694 views

What is this disintegration-like theorem?

This is cross-posted at MSE. I'm looking for a reference for the following result. It seems like it must be known, or follow quickly from something known, but I have not been able to find it in any ...
user435571's user avatar
1 vote
1 answer
80 views

What the probability of the max value of restricted random variable? [closed]

$x_1 + x_2 + \dots + x_n = 1, 0 \leq x_i \leq 1$, and $(x_1, x_2, \dots, x_n)$ evenly distributes on its restricted space, obviously which is a polygon on $n - 1$ dimension plane. Let random variable ...
Dandiss Valjean's user avatar
9 votes
1 answer
169 views

Bounding maximum probabilities in sum of i.i.d discrete RVs

Let $X$ be a a discrete RV with $\mathbb{P}(X=k)<p$ for every $k$ (that's all we know). Taking the independent sum $S=X_1+X_2+\cdots+X_n$, with each $X_i$ distributed like $X$, what can we say ...
Flo Pfender's user avatar
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2 votes
1 answer
120 views

Entropy of distribution with block matrix support

Let $P(X_1,X_2)$ be a discrete bivariate distribution that has the form shown in the figure below, i.e. its support can be split into blocks that do not overlap on either dimensions. Let's build $P'(...
Cesare's user avatar
  • 189
-1 votes
1 answer
306 views

expectation of upper quantile proportion

(edited considerably following comments) We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
Amit Portnoy's user avatar
1 vote
1 answer
135 views

algebraic tail of a random variable

Could anyone explain to me what does it mean by a map $f\to K_f$ and $f\to \rho(f(x_0), x_0)$ has an algebraic tail relative some measure, where $\rho$ is Prohorov metric, from this paper Paper which ...
Myshkin's user avatar
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3 votes
0 answers
117 views

Embedding a continuous-time martingale in Brownian motion

Using the Skorohod embedding, we can embed any square-integrable discrete time martingale $(M_n)$ into a Brownian motion, obtaining times $(T_n)$ such that $(B(T_n))_{n\ge 0}$ is a version of $(M_n)$. ...
Eric Foxall's user avatar
-1 votes
1 answer
81 views

Convergence in mean and convergence in distribution

Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that $$ 0< ...
Wenguang Zhao's user avatar
0 votes
1 answer
159 views

About another potential characterization of normal numbers

Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and ...
Vincent Granville's user avatar
0 votes
1 answer
957 views

Convergence in distribution of products

Suppose that a sequence of random variables $Y_n$ convergence in $L^2$ to $Y$, i.e. $$ E|Y_n-Y|^2\to0\quad \text{as}\quad n\to\infty. $$ Moreover, there exist constants $c_0$ and $c_1$ such that $$ 0 &...
Wenguang Zhao's user avatar
0 votes
0 answers
45 views

Skorohod Space with $J_1$ topology homeomorphic to Frechet Space

Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
ABIM's user avatar
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2 votes
1 answer
214 views

Reference: hitting time of Gaussian process

Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by $$ Y_t = y+\int_0^t X_s ds + W_t, $$ for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...
ABIM's user avatar
  • 5,019
1 vote
0 answers
63 views

Approximation of measured-valued function by continuous functions

For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e., $$ \int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty. $$ Let $\mu$ be a probability measure on $R^d$ such that $$ \int_{R^d}\int_{R^d}(|z|^2\...
Wenguang Zhao's user avatar
-1 votes
1 answer
115 views

Approximation of function in general measure space

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
Wenguang Zhao's user avatar

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