The convergence tag has no wiki summary.

**16**

votes

**1**answer

108 views

### Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series
$$f_{\sigma}(z) = ...

**-2**

votes

**0**answers

26 views

### Expectation of O_p(1) process [migrated]

Suppose $\{X_n \}$ is bounded in probability, i.e. $Prob(|X_n| > M_\epsilon) = \epsilon$ for all $n > N_\epsilon$, $M_\epsilon < \infty$.
Is there any condition(s) to guarantee that ...

**1**

vote

**1**answer

210 views

### Convergence of a Trigonometric Series

After working with a Fourier series for a while, I realized that it would be of great help to me if I could prove that the following limit is zero:
$$\lim_{N\to\infty}_{N\in ...

**0**

votes

**0**answers

58 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**0**

votes

**1**answer

113 views

### Is $\int_0^\infty \sin(Kx)f_K(x)\,dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\,dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...

**0**

votes

**1**answer

48 views

### Preserving Predimension Functions under Functional Convergences

Definition 1. If $\mathcal{L}$ is a countable relational language, a predimension class $C$ is a class of $\mathcal{L}$-structures with the following properties:
C1: ...

**1**

vote

**0**answers

36 views

### Epi-convergence to indicator function

Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...

**5**

votes

**1**answer

172 views

### Convergence rate of the central limit theorem near the center of the distribution

I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...

**2**

votes

**0**answers

37 views

### Convergence of Symmetric Iterative Proportional Fitting

Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by ...

**4**

votes

**2**answers

222 views

### Estimate on gaussian distribution

Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...

**0**

votes

**1**answer

115 views

### Some convergence similar to weak-$\ast$ convergence on the space of finite measures

I have a question:
Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$.
Define $TV(x)$ as the total variation ...

**1**

vote

**0**answers

57 views

### Berry-Esseen result for triangular arrays

Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ ...

**3**

votes

**0**answers

93 views

### Quantile convergence

Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote
$$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$
the empirical distribution function. Suppose that we know ...

**0**

votes

**1**answer

183 views

### Convergence of the empirical distribution function

Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a ...

**2**

votes

**0**answers

57 views

### Greedy interpolation of functions

Let $f:[-1,1]\rightarrow \mathbb{R}$ be a continuous function. Consider the following greedy algorithm for interpolation:
Set $r_0 = f$.
for $k = 0,1,\ldots,$
Find the location of the global ...

**7**

votes

**2**answers

289 views

### Convergence rate of the convolution of almost uniform measures on $\mathbb{Z}_p$

Statement Given a finite abelian group $G$ and two independent random variables $X,Y$ taking values in $G$ and satisfying $d_{TV}(X,U_G)\leqslant \delta$ and $d_{TV}(Y,U_G)\leqslant \delta$ (where ...

**0**

votes

**0**answers

72 views

### Absolute convergence of double Fourier-Haar series

Let $\left\{\chi_n\right\}_{n\geq 0}$ be a complete orthonormal (wrt a measure $\mu$) set on $[-1,1]$. Given a function $f:[-1,1]^2\rightarrow \mathbb{C}$, what smoothness requirements on $f$ are ...

**5**

votes

**2**answers

275 views

### How general is the convergence of the exponential function's power series?

I asked essentially this over two weeks ago on MSE, and nothing was else was posted to that question.
Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$.
Let $\;\; \beta \: : \: V\times ...

**5**

votes

**3**answers

625 views

### On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$.
I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...

**1**

vote

**0**answers

69 views

### Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method

I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...

**2**

votes

**0**answers

145 views

### An inequality for Lp-functions

I am interested in the following inequality:
\begin{equation}
\int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...

**-2**

votes

**1**answer

152 views

### non-trivial convergent sequence [duplicate]

I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...

**7**

votes

**1**answer

225 views

### Ultralimit versus partial limit

Let $\omega$ be a nonprincipal ultrafilter on $\mathbb N$.
A standard construction gives an $\omega$-limit, say $x_\omega$, for any bounded sequence $(x_n)$ of real numbers.
Namely, there is unique ...

**2**

votes

**3**answers

178 views

### Speed of convergence for Weyl's Equidistribution theorem

If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...

**1**

vote

**1**answer

169 views

### Convergence of a sum to the integral

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
...

**7**

votes

**1**answer

288 views

### What should the morphisms in the Category of Directed Sets be?

Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...

**-2**

votes

**1**answer

343 views

### weak star convergence [closed]

if $F$ is a Banach space and $f_n \subset F^* $ weak star convergent to $f\in F^*$. If further $x\in F$ is the weak limit of $(x_n)_n \subset F$ does then $f_n(x_n) \longrightarrow f(x)$ hold?
We ...

**1**

vote

**1**answer

235 views

### Pointwise convergence of double Fourier series

I'm looking for references/theorems that deal with the pointwise convergence of double Fourier series expansions for a particular function.
Let $D \subseteq [-\pi, +\pi]^2$ be an arbitrary set of ...

**0**

votes

**0**answers

45 views

### Is it possible to define a mixed normal having conditional variance almost everywhere null?

I'm trying to proving the stable limit of a martingale M_n(t).
When I calculate the limit in probability of its quadratic variation, I find that it
is always null except for a point.
It seems to me ...

**1**

vote

**0**answers

103 views

### Bounding a recursively defined sequence

I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined recursively as
$$ \lambda_0 = \frac{1}{2},$$
and
$$\lambda_{k+1} = \max_{\lambda\in [1,b]} \left(\frac{1}{2\lambda}\prod_{0\leq ...

**0**

votes

**1**answer

133 views

### splitting one limit into two?

Suppose I have the limit
$\lim_{m\rightarrow \infty}\frac{\sum_{k=0}^ma_{k,m}}{\sum_{k=0}^mb_{k,m}}$.
When can I write this as
$\lim_{n\rightarrow \infty}\lim_{m\rightarrow \infty} ...

**1**

vote

**2**answers

295 views

### Limit with theorem of dominated convergence

Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$)
I have to calculate this limit
$$\lim_{|x-y|\to ...

**4**

votes

**0**answers

149 views

### Well-founded families of sets and topological convergence

Background/Motivation
A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) ...

**0**

votes

**1**answer

132 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 ...

**0**

votes

**1**answer

185 views

### Convergence in norm of Sobolev spaces

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\lbrace{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\rbrace}$ and I have to show that the function ...

**1**

vote

**0**answers

90 views

### Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain

Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique ...

**4**

votes

**2**answers

362 views

### Central Limit Theorem (and Berry-Esseen theorem) for non-independent variables

Consider the triangular array $X_{n,k}$ such that, for each $n>0$, the variables $(X_{n,1},\cdots,X_{n,n})$ have the following properties:
For any given $1 \le L \le n$, all
subsets of
...

**2**

votes

**1**answer

136 views

### Central Limit Theorem for additive function of permutations of sequences

Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...

**3**

votes

**1**answer

117 views

### Statistical properties of principal components and their convergence rates.

Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present ...

**1**

vote

**0**answers

167 views

### convergence of sets and limit of an integral

Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets.
Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function.
Let $s:Y\rightarrow X$ be a function (not necessarily continuous).
...

**0**

votes

**1**answer

355 views

### Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?

Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...

**5**

votes

**1**answer

181 views

### Failure of the Pointwise Ergodic Theorem

It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences.
The counter-example usually given is the Folner sequence ...

**3**

votes

**1**answer

130 views

### Does martingale convergence hold for arbitrary time?

Let $\{\mathcal B_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal ...

**0**

votes

**0**answers

295 views

### Series with iterated log's: does it converge?

This came up in our office today. Let
$$f(x) = \begin{cases} x & \mbox{if } x\leq 1 \cr x\cdot f(\ln(x)) & \mbox{otherwise}\end{cases}$$
Does this series converge?
$$ \sum_{n=1}^\infty ...

**1**

vote

**1**answer

152 views

### On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...

**1**

vote

**0**answers

340 views

### Definition and Convergence of Iteratively Reweighted Least Squares

I've been using iteratively reweighted least squares (IRLS) to minimize functions of the following form,
$J(m) = \sum_{i=1}^{N} \rho \left(\left| x_i - m \right|\right)$
where $N$ is the number of ...

**0**

votes

**0**answers

220 views

### Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...

**0**

votes

**2**answers

235 views

### Convergence of Fourier series for $C^p$ functions

Let $f \in C^p[0,2\pi]$ and periodic. Denote $\omega_p$ as the moduli of continuous of $f^{(p)}$. Then
$
|f - S_Nf| \le K \frac{\log{N}}{N^p}\omega_p(2\pi/N),
$
where $S_N$ is the Fourier partial sum ...

**1**

vote

**1**answer

174 views

### Convergence of Difference Quotients

Let $\gamma_{\varepsilon} \rightharpoonup \gamma$ in $W^{1,\infty}(0,1)$. Then for any fixed $s \in \mathbb (0,1)$ does the limit $\lim_{\varepsilon \rightarrow 0} ...

**0**

votes

**2**answers

475 views

### Weak versus strong convergence

This is my first time posting.
I am well aware that an $L^2$ weakly converging sequence is not convergent in the corresponding strong topology. However, my question is as follows, do the sequence of ...