The convergence tag has no wiki summary.

**38**

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**1**answer

745 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) = ...

**9**

votes

**1**answer

335 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 ...

**8**

votes

**2**answers

2k views

### Are weak and strong convergence of sequences not equivalent?

For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have ...

**7**

votes

**1**answer

316 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 $p \preceq r$ ...

**7**

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**2**answers

321 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 ...

**6**

votes

**3**answers

294 views

### Natural topologies for the space of rational functions

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree ...

**6**

votes

**2**answers

235 views

### What good is (strong) mixing in dynamical systems?

For measure-preserving dynamical systems, there exist several notions of mixing. The most basic ones are strong mixing, weak mixing and ergodicity (see the wikipedia page, for instance), asserting ...

**6**

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**2**answers

1k 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 ...

**6**

votes

**1**answer

76 views

### Summability of ratios of moments a weight

Recently, I encounter the following problem:
Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e.,
$$m_k=\int_0^1t^kw(t)dt.$$
Under what condition can we have
...

**5**

votes

**1**answer

1k 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 ...

**5**

votes

**1**answer

253 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 ...

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**2**answers

344 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

**1**answer

130 views

### Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...

**4**

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**2**answers

521 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
...

**4**

votes

**2**answers

250 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 ...

**4**

votes

**1**answer

119 views

### Weak convergence in $W^{1,p}_0$

Note from the answerer : this question stems from this article.
I ask this question in http://math.stackexchange.com/questions/1206617
I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...

**4**

votes

**1**answer

82 views

### Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.
Let $a_k >0$ be an increasing sequence ...

**4**

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**0**answers

189 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) ...

**3**

votes

**1**answer

143 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 ...

**3**

votes

**1**answer

43 views

### Error for the convergence by distribution

A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = ...

**3**

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**1**answer

161 views

### On the domain of convergence of usual definition of Riemann zeta function [closed]

If we define Riemann zeta function as it is usually defined so that $$\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}e^{-s\ln n}$$ then we can rewrite it as:
$$\zeta ...

**3**

votes

**2**answers

141 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 ...

**3**

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**1**answer

271 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...

**3**

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**0**answers

161 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 ...

**2**

votes

**3**answers

237 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 ...

**2**

votes

**1**answer

214 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 ...

**2**

votes

**1**answer

1k views

### Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...

**2**

votes

**2**answers

138 views

### Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent (“weakly”)?

It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I ...

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**1**answer

70 views

### Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...

**2**

votes

**1**answer

108 views

### Growth of the truncation of the integral multiples of an irrational number

Let $[a]$ denote the integral part of a real number $a$.
Let $a$ be an irrational number and $b$ a real number greater than $1$.
Consider the sequence $(b^n(na-[na]))$ with $n$ running on the ...

**2**

votes

**1**answer

176 views

### Linear or quadratic combinations of i.i.d. random variables [closed]

I already posted this question here http://math.stackexchange.com/questions/769920/law-of-large-numbers-for-linear-quadratic-combinations-of-i-i-d-random-variab but I received no answers.
Let ...

**2**

votes

**1**answer

179 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 ...

**2**

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**0**answers

38 views

### Decomposition of the space according to the Ergodic Theorem

Given a space $(X, T)$, it is well known that for every $T$-invariant ergodic measure $\mu$, there exists a set $E_\mu$ of $\mu$-measure $1$ s.t. for every "nice" function $f$
$$ ...

**2**

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**1**answer

74 views

### Convergence of Fixed-Point Iteration of a dependent map

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore ...

**2**

votes

**1**answer

101 views

### Variant of Skorokhod's theorem

Consider the following situation:
$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
There is a a random variable $\zeta: \Omega \to S$.
$f_n(\zeta) \to^d \eta$, ...

**2**

votes

**0**answers

97 views

### On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$):
$$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...

**2**

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**0**answers

112 views

### Speed of Approach to Invariant Measure

Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, ...

**2**

votes

**1**answer

250 views

### Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets

Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a finite subset of ...

**2**

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**0**answers

98 views

### Pointwise convergence of ergodic averages of unconventional conditional expectations

Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ...

**2**

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**0**answers

72 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 ...

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**0**answers

84 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 ...

**2**

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**0**answers

157 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 + ...

**1**

vote

**1**answer

188 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 :
...

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vote

**1**answer

196 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 ...

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vote

**2**answers

531 views

### Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left ...

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vote

**1**answer

222 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} ...

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vote

**1**answer

79 views

### Division methods for divergent continued fractions

I hadn't even noticed before entering the subject above the parellel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...

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vote

**1**answer

227 views

### Book on Convergence Concepts in Probability without Measure Theory [closed]

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...

**1**

vote

**1**answer

213 views

### Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...

**1**

vote

**1**answer

271 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 ...