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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Representation of support of Gaussian measure by kernels of no-variance functionals
Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for
$$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$
...
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Equivalent Definitions of the Gaussian Surface Measure for Regular Sets
I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
- 1,127
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Lévy measure and Lévy process
A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$
A Lévy process can be characterized by triples $(b, A, \...
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Is there an easy way to convert a non-deterministic optimal policy to a deterministic optimal policy for a given MDP?
For a MDP (Markov Decision Process) is there an easy way to convert a non-deterministic optimal policy into a deterministic optimal policy?
The trivial way will take $O(|\mathcal{A}|^{|\mathcal{S}|}$...
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Solving recursion / finding generating function of a probability mass function
I am assessing the probability distribution on a running time of some algorithm that we've developed. I am looking for a family of probability mass functions $f_n$ with the following recurrence:
$$
f_{...
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Question abouth Skorokhod representation of random variables
It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...
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Hitting time of an Ornstein-Ulhenbeck process
If we consider a nice Ornstein-Uhlenbeck process
$d x (t) = - \gamma x(t) \,dt + \sigma \,d w (t)$
with $x(0) = x_0 \in (-L,L)$.
Here $\gamma, \sigma$ are positive constants and $w(t)$ is a Wiener ...
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2
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Number of samples needed as input to Bernoulli factory
Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...
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Are most random variables trivially sub-gaussian? [closed]
I'm trying to understand sub-gaussian RVs to see if they could be relevant to my work.
The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform /...
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Gradient of probability distribution
Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ ...
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Sub-$\sigma$-algebras and conditional expectation
Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space?
In other words, is it true that conditional ...
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Rough path theory- Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,r} \otimes dX_r$
This is exercise 7.7 from Martin Hairer's Rough Path notes.
Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,r} \otimes dX_r$ where the integral is to be interpreted in the sense of (4.22) (I'll define ...
user69208
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Regarding left-to-right minima
Let $\rho$ be a permutation on $[1,n]$ and $l_i$ be the number of left-to-right minima in $\rho_{i\ldots n}$, I know that for a random permutation $E[l_1] = H_n$ (the $n$-th Harmonic number) but is ...
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Picking codewords that are close
I posted this question in https://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...
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Random non-intersecting circles in the plane
If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
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Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations
Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.
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Strictly positive solutions of a random linear system
Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, ...
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An identity for the exponential of a martingale
I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...
3
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2
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546
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Measure concentration for law of large numbers
The classical law of large numbers states that
$$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$
for i.i.d. $X_1, X_2, \ldots$ with finite $L^1$ norm.
I was wondering whether is it possible to ...
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Ratio of expected diameter and height of a conditioned Galton-Watson tree
A Galton-Watson tree is the family tree of a Galton-Watson process. Let $T_n$ denote a Galton-Watson tree conditioned on total population size $n$. The time of extinction is its height $H(T_n)$ and ...
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Sampling from maximally skewed stable distribution
I am reading a paper which refers to a maximally skewed stable distribution $F(x;1,-1,\pi/2,0)$ . Is there an efficient way to sample from this distribution?
If $X$ has distribution $F(x;\alpha,\...
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Does $Mv$ converge to i.i.d in some sense?
I am not a professional mathematician so please excuse me if my question is not phrased correctly.
I am interested in the following simple sounding problem.
Consider a random $n$ by $n$ $0$-$1$ ...
- 3,195
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Characterization of a set in $\mathbb{R}^d$
Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...
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Random weighted selection without replacement
I am using the following procedure to select $m$ different numbers $\{i_1,\ldots,i_m\}$ from the set $\Omega = \{1,\ldots,N\}$, with $m,N\in\mathbb{N}$ such that $m< N$.
Selection procedure
...
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How fast does the Heat equation with boundary condition $\frac{\partial u}{\partial \vec{n}}=u^2$ decay?
Consider the heat equation $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u$ in a bounded domain (say the interval [0,$\pi$]) with boundary condition $$\frac{\partial u}{\partial \vec{n}}=u^2$$
with ...
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A question about stochastic kernels and invariant measures
Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times ...
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The pdf of of the determinant of an interesting random matrix
Consider a matrix of order $n$ whose elements are independently and uniformly chosen from the interval $[a,b]$ wher both $a$ and $b$ are positive .Let $X$ be the r.v. corresponding to the determinant ...
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Proving that Brownian motion has no points of increase
I am reading Burdzy's paper on the points of increase of Brownian motion:
Burdzy's Paper
He is proving that, almost surely, a Brownian motion, has no points of increase. What he actually proves is ...
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Uniform bound on the rate of convergence of the renewal measure
Consider a renewal process whose holding times are given by a continuous random variable $X$ supported on $[0,1]$. It is known (e.g. Stone '65) that the renewal function $m(t)$ converges to $t/\mathbb{...
- 1,886
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Extending Tarski's Theorem on invariant measures
Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am ...
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Removing edges from Erdős–Rényi graph to make two nodes disconnected
Consider a Erdős–Rényi graph on $n$ nodes, say $1,2,\ldots,n$, ($n\geq 3$) such that the probability of edge between any two nodes is $c/n$. I wish to know if there is a result that says
"There ...
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Chernoff-Hoeffding bound for complex values
Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value
$\mu$ and satisfying $|X_i| \le b$.
Let $\epsilon > 0$. ...
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Probability Density Optimization
I am working on an optimization problem which I am stuck on towards the end.
Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x,y)$, now I define ...
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Upper bound concerning Snell envelope
Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
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Independence using reflecting brownian motion
Suppose $X$ and $Y$ are two Brownian motions such that $|X|$ and $|Y|$ are independent. Then it is easy to show that $\langle X,Y \rangle =0$ using the Tanaka formula, for example, and thus $X$ and $Y$...
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Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
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Large deviations for sums of exponentially distributed random variables.
Take a large integer $R$, and let $(X_j)_{j\geq R}$ be a sequence of exponentially distributed random variables with parameters $\pi_j := j^{1+\alpha}$ ($\alpha>0$), so that $\sum_{j\geq R} \frac{1}...
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Bounds for duplicate finding with limited independence
(This is a follow up to this previous question on math.stackexchange.com.)
Assume a process that samples uniformly at random from the range $[1,\ldots,n]$. I am interested in the time to find a ...
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2
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Itô's Formula on a bounded Domain
Let $U$ be a connected and bounded Domain, w.l.o.g. we choose $[0,1]^2$ and let $f \in \mathcal{C}^2((0,1)^2)$ with $\Delta f(x)=0$ for $x \in (0,1)^2$ and having normal derivative of $0$ almost ...
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501
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A probability question about removing stones from piles
I have run across a question that seems like it should have a well known answer, but I can't find one, so I thought I would ask this hive mind:
Suppose we start with t piles of s rocks each. In a ...
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Continuity of hitting distributions
Hi everybody
Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...
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"Generalized" monotonicity of the expected value
Let $X$ and $Y$ be random variables
with cumulative distribution functions $F_X(t)$ and $F_Y(t)$ respectively.
Suppose that
$\forall t \in \mathbb{R} \ F_X(t) \geq F_Y(t)$.
Does it imply that $E(X) \...
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Bounds on tails with moments
A sort of continuation of Comparing distributions with moments
Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\...
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Infima of conditional densities after disintegration
Consider the measurable partition of the open unit square $(0,1)\times(0,1)$ into horizontal intervals $L_y=(0,1)\times\{y\}$. Let $\mu$ be a Borel probability measure with the disintegration
$$
\mu(...
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195
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bounding the probability that a polynomial is near 0
Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
- 4,354
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Convergence of stopped Brownian motion
Suppose $B$, $B_n$ are Brownian motions, and write $B^s$ for $B$ stopped at the first time equals $k$, say. (Similarly $B^s_n$).
I know how to prove the following: if $B_n \to B$ uniformly on ...
- 2,775
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1
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238
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Bounding the success time of a coupon collector like problem
Consider the complete graph on $n$ vertices. Each step, one chooses one of the $\binom{n}{2}$ edges iid uniformly at random. Say a sequence of choice is successful if there is some permutation of the ...
- 4,354
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1
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Standard way of determining if you have enough data to reliably compute success probability
Given $s$ successes in $n$ trials, where $p=\frac{s}{n}$, is there a standard way to determine if I have enough data to compute a meaningful statistic? For example, given $s=1, n=10, p=0.1$, the 95% ...
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mutual hitting measure between two sets
Given disjoint nonempty subsets $X_1, X_2$ of the state space of a finite irreducible Markov chain, there are unique measures $\mu_1$ on $X_1$ and $\mu_2$ on $X_2$ such that (a) starting from a $\mu_1$...
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Concentration of measure and bounds on variance
I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao ...
