All Questions
9,497 questions
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444
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How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
0
votes
0
answers
582
views
When an integral with respect to a Poisson point process is finite?
Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ \...
- 724
0
votes
0
answers
145
views
Discrete measures and discrete kernels
This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
- 215
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0
answers
216
views
Computation on Random Bipartite graphs
I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
- 903
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votes
0
answers
133
views
What is the sigma field of the derivative of a process?
When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
- 19
0
votes
0
answers
42
views
Probability of close approach for multivariate normal variables
The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
- 195
0
votes
1
answer
208
views
Lower bound for median of independent outcomes
Consider a stochastic variable $X$ taking positive real values and the events $P(X\geq a)\leq\frac{1}{3}$ and $P(X \leq b) \leq \frac{1}{2.9}$. We define $X_m$ as the median of $k$ independent ...
- 75
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0
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454
views
Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
0
votes
0
answers
57
views
Numerical method for self-consistency of one-dimensional probability density function
I have an integral equation for self-consistency of one-dimensional probability density function, like this
$$\rho_x(x) = \frac{1}{|a|}\int \int \rho_x\left(\frac{s-b}{a}\right) \rho_P(p) \delta(x-g(...
- 111
0
votes
1
answer
113
views
Expected number of packed items in box
Assume we have a box of size $n$, some items $X_i, i \in N$ of unknown distribution, with expected size $\mu>0$ and variance $\sigma^2$. We want to randomly and greedily pack the box with the items ...
- 75
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0
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161
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question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
- 1,835
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votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
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0
answers
232
views
Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
- 585
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1
answer
215
views
Residual lifetime of heavy-tailed random variable
The residual life time distribution of a random variable $X$ with distribution function $F$ is given by the formula
\begin{equation}R(t)=P[X_\text{res}\leq t] = 1-\frac{1}{\mathbb{E}[X]}\int_{y=0}^\...
- 230
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votes
0
answers
184
views
Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
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0
answers
94
views
Dominating Poisson with parameter depending on a Bernoulli
Fix $\mu >0$ and take $\lambda \geq 0$. Let $B_p \sim \text{Ber}(p)$ with $p = \exp(-\mu - \frac{\lambda}2) $. Define the random variable $Y$ which is Poisson with parameter depending on the value ...
- 191
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0
answers
204
views
Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators
Consider the operator on $\ell^2(\mathbb{Z})$
$$
H = \Delta + v.
$$ Here $\Delta$ is the nearest neighbour Laplacian on $\mathbb{Z}$, $\Delta_{k, \ell} =1 $ if $|k - \ell| =1 $ and zero otherwise, ...
- 195
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0
answers
134
views
Mittag-Leffler function and Laplace Integral
Let $E_{\alpha}(z)\triangleq \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + 1)}$ be the Mittag-Leffler function.
I am looking for a full proof of the following fact (a reference to a proof in the ...
- 424
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0
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917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
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0
answers
257
views
Sum over a product of binomial coefficients related to a collision problem
I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:
$$R\left(n,m,j\right)=\sum_{k=0}^{n}...
- 1
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votes
1
answer
408
views
Generating independent random variable from two correlated random variables
Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
- 1,109
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0
answers
522
views
How to simplify conditional probability of union of several events
I have an output binary scalar, $y∈B=\{0,1\}$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=\{0,1\}$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...
- 133
0
votes
0
answers
536
views
matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
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votes
0
answers
404
views
When is the median closest nearest-neighbor distance larger than the mean closest nearest-neighbor distance?
Consider a random Poisson process in an $d$-dimensional cube of arbitrary size (alternatively, consider an arbitrarily large $(d-1)$-dimensional sphere in an $d$-dimensional space). If we have a ...
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0
answers
208
views
Brownian particles in a box: the probability that a sphere (of some radius) centered on a particle only contains one particle for a duration of time
Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a ...
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0
answers
104
views
Minimum distance larger than a fraction $f$ of the closest nearest-neighbor distances for points placed by a random Poisson process?
Consider a random Poisson process on arbitrarily large volume in $R^d$ enclosed by an $R^{(d-1)}$ dimensional sphere. The process terminates when a density of points $\rho$ is achieved (letting $N$ ...
- 1
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0
answers
213
views
Behavior of the sum of the exponents of chi-squared random variables normalized by their maximum
Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ ...
- 907
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votes
1
answer
184
views
Can I obtain the limit value of a linear spectral statistics using Stieltjes transform?
I would like to calculate the limit value of a linear functional
\begin{equation}
\lim_{n\rightarrow\infty}\mathcal{I}_n=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n f(\lambda_i)=\lim_{n\...
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votes
0
answers
145
views
multivariate integral calculation in closed form
I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
- 109
0
votes
0
answers
151
views
Inequality relating stationary probabilities and transition probabilities
Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains ...
- 23
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votes
0
answers
1k
views
Measure induced by function
It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also 2-...
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votes
0
answers
68
views
probability P that all circles are connected with each other.
Let N circles with homogeneous radius r are deployed with Poisson distribution in area A. These circles are connected if there euclidean distance is less than r.what is the probability P that all ...
- 1
0
votes
0
answers
96
views
Finding conditions on unspecified CDF that permit a solution to an equation
[A duplicate thread can also be found at
https://stats.stackexchange.com/questions/59450/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ]
Let $F(\alpha) := \mathbb{P}(\...
- 1
0
votes
0
answers
104
views
Big eigenvalues of a special stochastic matrix
Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq |\lambda_2|(M)....
- 105
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0
answers
347
views
An interesting version of the problem “balls into bins”
Consider n people, each has k identical balls. Each people choose k different bins from m bins, constrained by the condition that there are no two people choose exactly the same k bins. For instance, ...
- 1
0
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0
answers
91
views
Pruning copies of an element from a multiset via a uniform random selection process - does vigilance matter?
This is an extension of a previous question of mine (nicely answered by Douglas Zare): Filling a bin with one type of element when uniformly selecting from a set of two (with bias)
Say I fill a ...
- 43
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votes
0
answers
118
views
mathematical expectation of length of dependency well.
We have these assumptions:
$W$ is a finite set
$\mathcal W$ is the set of all functions $f:W\to \mathcal P(W)$.
$p:\mathcal P(\mathcal W)\to [0,1]$ is a probability measure.
For each $w\in W$ and $m\...
- 63
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votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
0
votes
0
answers
160
views
Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
- 103
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0
answers
102
views
Efficient algorithm for computing the mixed moments of sums of random variables
Let $X_1,\dots,X_m$ be dependent random variables. We are interested in efficient algorithms for computing the following quantity:
$$E\Big[\Big(\sum_{i=1}^m X_i\Big)^k\Big],$$
where $k\in\mathbb{N}$ ...
- 1
0
votes
0
answers
134
views
Hausdorff distance and sum of independent variables
Consider a probability space $(\Omega, \mathcal{F}, P)$, as well as two sub-$\sigma$-fields $\mathcal{A}$ and $\mathcal{B}$. The Hausdorff pseudo-distance between $\mathcal{A}$ and $\mathcal{B}$ is ...
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0
answers
655
views
Extension of probability measure from a finite algebra to sigma-algebra with countable many generators
I apologize for probably trivial question, I am far from this field.
If $\mathcal A$ is a $\sigma$-algebra of subsets of $X$ (for example Borel sets of Cantor space $2^\omega$), can I extend to $\...
- 1
0
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0
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112
views
Markov renewal process with failure?
I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail.
I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = (...
- 101
0
votes
1
answer
2k
views
Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral
Hi, I have the following expected value to compute
$E[ \int_{o}^{T} f(t) dt \int_{o}^{T} H(s) dW(s)]$,
where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the ...
- 47
0
votes
1
answer
129
views
Probability of summing products of irreducible polynomials in a finite field to zero
Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...
- 3
0
votes
0
answers
155
views
Convexity of a Certain Set of Covariance Matrices
Hello,
My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
- 155
0
votes
1
answer
124
views
Is there a known asymptotic scaling for the probability of recurrence for a walk on $Z^d$?
I'm curious if there is a known asymptotic scaling for the return-to-origin (i.e. recurrence) probability for a random on $Z^d$ as a function of $d$?
Mathworld gives the recurrence probability:
...
0
votes
0
answers
179
views
Arithmetic properties of erf functions
I was messing around with Benford's law trying a proof to fill up time on a Saturday, and I ran into a problem. I have the equation $\frac{\mathrm{erf}(2x)-\mathrm{erf}(x)}{\mathrm{erf}(10x)-\mathrm{...
- 1
0
votes
0
answers
166
views
Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?
Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
- 87
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