All Questions
18,180 questions
1
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141
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$A \perp B$ and $A+B\perp r\left( 2A+B\right)$ for some continuous function $r$. Is there such a triplet $\left( A,B,r\right) $ with non-constant function $r$?
Let $A$ and $B$ be independent continuous random variables with supports $ \left( -\infty ,\infty \right) $ and $r$ be a continuous function. In addition, $A+B$ and $r\left( 2A+B\right)$ are ...
- 11
0
votes
0
answers
293
views
Open Jackson network with deterministic arrivals.
Dear Friends,
Is there any known Jackson-like theorem for an open Jackson network with deterministic arrivals?
Thanks,
Michael.
- 85
2
votes
0
answers
102
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More positive pivotal edges than negative ones at critical bond percolation on Z^2?
Consider critical bond percolation on $\mathbb{Z}^2$ inside a fixed rectangle $(0,0) - (an,n), a \geq 1$ and write $A$ for the event that there is an open crossing in the long (left to right) ...
- 960
3
votes
1
answer
280
views
An analogue of an old proposition
For the absolute value $|C|=(C^*C)^\frac{1}{2}$ and the
Hilbert-Schmidt norm
$\parallel C\parallel_{HS}=(trC^*C)^\frac{1}{2}$ of the operator $C$. The
following inequality is shown by Araki et al in ...
- 223
4
votes
0
answers
497
views
A Local CLT with large variance
For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a
random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
- 263
3
votes
0
answers
359
views
Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs
$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems,
Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$
and a large set of initial ...
- 2,044
1
vote
0
answers
135
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Inequality involving BV norm and a regularizing kernel
In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ...
- 2,222
3
votes
1
answer
122
views
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% ...
- 133
1
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0
answers
159
views
variational problem under convexity constraints
I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert ...
- 1,839
4
votes
1
answer
228
views
When can closedness of the range of an operator be checked on a positive cone?
Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements ...
user2412
0
votes
0
answers
320
views
A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
2
votes
1
answer
186
views
scalar diffusions are reversible
It is well known that under mild assumptions a scalar diffusion $dX_t = a(X_t) dt + \sigma(X_t) dW_t$ with invariant probability distribution $\pi$ is reversible. This is indeed not true for ...
- 2,133
6
votes
0
answers
299
views
Spectrum of an operator arising in a dynamical problem
(Question edited according to Denis Serre comment).
While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(\mu)\...
- 14.4k
3
votes
0
answers
125
views
Is a parametric family which is universally consistent for multiple quantiles impossible?
Suppose I am dead-set on using Bayesian inference on independent and identically distributed data, but I'm lazy and insist on using a parametric likelihood function come what may. I'd be reassured to ...
- 2,791
0
votes
0
answers
184
views
Integration of discounted normal distribution
Hi
I want to find expectation of integration of normal distribution $\varphi(t)\sim N(0,\sigma\sqrt t)$
but i also want to discount it continuously with parameter $\alpha$.I mean i need to ...
- 1
5
votes
0
answers
308
views
Is the nearest walk to Brownian motion approximately uniform?
This is a follow-up to an earlier MO question.
Let $W:[0,1]\rightarrow\mathbb R$ be standard Brownian motion with $W(0)=0$.
Let $F_n$ denote the collection of all the $2^n$ many piecewise linear ...
- 24.8k
0
votes
0
answers
127
views
A problem about partial sum of random number composition
Consider the strong random number composition,
$x_1 + x_2 + \cdots + x_n = m$, with $x_i > 0$ and all possible compositions have the same probability.
Let random variable $S_i = \sum_{j=1}^i x_j$...
- 177
1
vote
1
answer
113
views
What is the probability that all numbers in a set P are unique and each number in P is chosen randomly between 1 and n^3? [closed]
Hope someone can help me answer this question.
The problem is described as below.
I want to form a set (P) of n numbers. I randomly choose a number between 1 and n^3 and I choose n times.
My ...
- 11
1
vote
0
answers
237
views
Variation of a function
There are probably some of you guys who already know some of the terms that I am going to use so in order to be not so boring I will put the definition to the end.
Let $f$ be a piecewise expanding ...
- 11
3
votes
1
answer
181
views
Reference request - spectral radius formula for linear transformations in char p
I am finishing up a paper and I would like to be able to quote a theorem that does what
is said in the title. To be specific let me introduce some notations:
${\bf F}$ is a local field of ...
4
votes
0
answers
189
views
Boundedness criterion for operators on mixed Lebesgue spaces
Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences
${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that
...
- 979
1
vote
0
answers
265
views
"Lift and project" procedure for matrices
Definition. Let us call $n\times n$ matrix with non-negative entries good if sum of every row and column is equal to $1/n$.
Suppose we have a good matrix $A$. Let us consider the following strange "...
- 1,791
-3
votes
1
answer
318
views
Porbability of selecting balls from boxes [closed]
There are three boxes. B1, B2, B3 The probability of selecting them is 0.2, 0.2 , 0.6 respectively.
B1 contains 3 red balls and 7 green balls.
B2 contains 5 red balls and 5 green balls.
B3 contains ...
0
votes
0
answers
138
views
Notion of simplicity of a function(al)
Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).
Specifically, intuitively one could ...
- 59
-1
votes
3
answers
304
views
Distribution under operations
Let $X$, $Y$, $Z$, and $W$ be i.i.d. copies of a standard gaussian variable, that is in distribution $\mathcal{N}\left(0,1\right)$, then what is the distribution of $\left|\frac{XY}{Z}-W\right|$?
...
- 65
2
votes
0
answers
240
views
Radon transform and Log-concavity
This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat f(\omega,t)$...
- 52.3k
3
votes
1
answer
145
views
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$...
- 19.7k
-1
votes
1
answer
311
views
A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\...
- 1
0
votes
0
answers
98
views
coupling of projections and projection of the coupling
Let $C$ be a coupling between two measures, $C= \mu^1 \mbox{ } t \mbox{ } \mu^2$ ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both ...
- 295
4
votes
1
answer
321
views
What functorial topologies are there on the space of linear maps between LCTVS?
Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-...
- 26.8k
2
votes
1
answer
493
views
Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
4
votes
0
answers
257
views
A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.7k
1
vote
3
answers
246
views
Extreme value theory
I have been asked to provide an "approximation at infinity" of an expression that at the end simplifies to $-\frac{b e^{-a t}-a e^{-b t}}{a-b}$, in a course about extreme value theory.
In the course, ...
- 19
3
votes
0
answers
134
views
SOS model - height
Let $\Lambda_n = \{ -n,\ldots,n\}^2$ and let $\{X_i\}_{i\in \Lambda_n}$ be the family of $\mathbb{R}$-valued of random variables with the density proportional to
$\exp(-\sum_{i\sim j} |X_i - X_j|),$
...
- 1,491
1
vote
1
answer
257
views
Two-Dimensional Gobbling Algorithm
Let (m,n) be an ordered pair of positive integers. While m>0 and n>0, let k_1 be a random positive integer between
1 and m and k_2 a random positive integer between 1 and n. Output (k_1,k_2). Let ...
- 897
4
votes
1
answer
363
views
Efficiently sampling points from an integer lattice.
Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...
- 41
1
vote
0
answers
323
views
Law of the sum of order statistics through MCMC
Hi everyone,
I was wondering is there were some known applications of MCMC methods for simulating the law of the sum ( or alternatively the joint law) of the order statistics of a multinomial (...
- 1,334
1
vote
0
answers
61
views
Distribution for probability of an incorrect inference based on a comparison of only two samples?
I'm trying to demonstrate the problems of how taking a sample and assuming it reflects the population accurately can be problematic.
Imagine say an urn with some large number of balls, black and ...
- 11
3
votes
0
answers
179
views
How can the topological entropy and $L^2$ mixing rate be related?
For a product of otherwise identical systems evolving at different rates, the toplogical entropy and a quantity very closely related to (indeed, identifiable with a nondegenerate variant of) the $L^2$ ...
- 15.4k
1
vote
0
answers
299
views
Markov Chain Patterns
Hi
I would like to detect repetitive patterns and deviations from these repetitions. I have historical data and can calculate probabilities for the transitions between my many states. I have ...
- 11
3
votes
1
answer
320
views
Joint Law with 2 marginals and marginal of the spread
I have a question for you and thank you in advance for your answers and ideas.
Let us suppose that we have the marginal distributions of two r.v X and Y, and also the law of X-Y (or any linear ...
- 375
2
votes
1
answer
321
views
How to show that an infinite sequence is normal if and only if every block of equal length appears with equal frequency?
An infinite sequence is normal if all strings of equal length occur with equal asymptotic frequency.
Formally, let $\Sigma$ be a finite alphabet of $b$ digits. Let $S$ be an infinite sequence and $\...
- 537
1
vote
0
answers
128
views
Proving that an optimal solution "converges"
This question is a follow-up on a previous question I asked at:
Distances between and among points in a region
Let $X = x_1,\dots,x_n$ denote a finite set of $n$ points in the unit circle $C$ in the ...
- 195
-1
votes
1
answer
129
views
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?
- 1
2
votes
0
answers
137
views
Invariant linear manifolds for multiplication by the independent variable in L^2 (R)
In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-...
- 31
0
votes
1
answer
137
views
Mean of an experiment
Suppose we have a bag of n different balls, and each time m (m<n) balls are taken out for checking from the bag and put back. ...
- 9
2
votes
0
answers
313
views
Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk
Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...
- 599
5
votes
0
answers
417
views
Direct integrals and fields of operators
Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.
Suppose now ...
- 3,897
1
vote
0
answers
169
views
Marginals and Convex Sets
I am looking for a weak convergence of marginals result, in the following type of situation. References to 'related' situations are also very much appreciated.
I have a collection of affine ...
- 105
3
votes
1
answer
366
views
Random generation of subsets using conditional probabilities
Edit: Rewritten with motivation, and hopefully more clarity.
I'm building a site for a card game called dominion. In it, people build 'decks' of 10 distinct cards from a set of (currently) ...
- 203