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Questions tagged [limits-and-convergence]

Convergence of series, sequences and functions and different modes of convergence.

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Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations

Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
Grandes Jorasses's user avatar
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Asymptotic properties of Schauder bases in Hilbert spaces

Let $\ell^2$ be the Hilbert space of square-summable complex sequences. Let $(e_m)_{m \in \mathbb{N}}$ denote the canonical basis of $\ell^2$. Let $(u1_m)_{m \in \mathbb{N}}$ be a Schauder basis of $\...
Matey Math's user avatar
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1 answer
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Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?

I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
AY Wer's user avatar
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Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$

Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to ...
PDEprobabilist's user avatar
9 votes
1 answer
486 views

Does the sequence formed by Intersecting angle bisector in a pentagon converge?

I asked this question on MSE here. Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
pie's user avatar
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1 answer
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Rate of convergence of mollified functions in $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
Akira's user avatar
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Convergence of mollified functions in weighted $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
Akira's user avatar
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6 votes
2 answers
317 views

Maximal eigenvalue of a real symmetric Toeplitz matrix

The $n×n$ matrix $A_n$ is defined by the elements $a_{ij}=n−|i−j|$. \begin{bmatrix} n & n-1 & n-2 & \cdots & 1\\ n-1 & n & n-1 & \cdots & 2\\ n-2 & n-1 & n &...
Drophet's user avatar
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Eigenvalues of N×N correlation matrices as N tends to infinity

I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero. ...
Zywoo_biu's user avatar
2 votes
1 answer
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Almost sure convergence of double averages of IID random variables

Let $ \{X_i\}_{i=1}^{P} $ and $ \{Y_j\}_{j=1}^{Q} $ be two sequences of independent and identically distributed (i.i.d.) random variables. $X_i$ and $Y_j$ are independent between all pairs of $i$ and $...
CWC's user avatar
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17 votes
2 answers
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Is it known that the the sequence 7n+1 diverges to infinity starting with 7?

In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
pie's user avatar
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1 answer
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Sufficient condition for uniform convergence of the Stieltjes transform

Let $\mu$ be a probability measure and $\mu_N$ be a sequence of probability measures. For simplicity we may assume them to have compact supports contained in $[-1,1]$. Define $$G_\mu(z):=\int\frac{\mu(...
Jiyuan Zhang's user avatar
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1 answer
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Limiting problem for variable being part of derivative of natural degree

I discovered some interesting behaviour of Riemman's functional equation, such assuming Ramanujan's summation; $$ \begin {split} \zeta(s) & = 2(2 \pi)^{s-1}\left(\frac{\pi s }{2}\right)\zeta(1-s)\...
Wreior's user avatar
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1 answer
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Convergence in probability of sample covariance for permutation invariant triangular arrays

Take two triangular arrays $X_{N,i}$ and $Y_{N,i}$ of random variables where $1 \le i \le N$. Suppose that the families $\{X_{N,i}\}$ and $\{Y_{N,i}\}$ are independent, and that the following ...
Greg Zitelli's user avatar
1 vote
0 answers
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Uniform distribution as argument for copula likelihood

I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
Grigori's user avatar
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9 votes
1 answer
780 views

Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

I asked this question on MSE here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
pie's user avatar
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2 votes
0 answers
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Power series expansions and limits of knot invariants

This question is moved from math stackexchange which I posted several days ago without an answer. Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type ...
Eric Ley's user avatar
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Does $\sum_{k = 0}^{qN} [e^{-\frac{\pi^2 k^2}{N}} - (\cos \frac{\pi k}{2 q N})^{8 q^2 N} ] \rightarrow 0$ as $q\rightarrow \infty$?

Let $N, q \in {\mathbb N}$. Let $S(N,q) = \sum_{k = 0}^{qN} [e^{-\frac{\pi^2 k^2}{N}} - (\cos \frac{\pi k}{2 q N})^{8 q^2 N} ] $. $N$ is fixed. Does $S(q,N)$ tend to zero as $q$ tends to infinity? How ...
fdowker's user avatar
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Does $\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]$ have a closed form?

In my MSE question, "Conjectured connection between $e$ and $\pi$ in a semidisk", the answer included $$\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]\approx 0.96454\ldots.$...
Dan's user avatar
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How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin

Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
Rodrigo's user avatar
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How to prove the convergence of Gechberg-Saxton algorithm?

I just have a problem that Gerchberg-Saxton algortihm is no worse than the previous iteration but not sure whether it is convergent.
Jianqing Li's user avatar
2 votes
1 answer
122 views

Convergence of the product of three sequences

Let $Q=(0,T)\times \Omega$, $\Omega$ being a bounded subset of $\mathbb R^d$, sufficiently smooth. Consider three sequences $ u_n$, $ v_n$, and $w_n$ such that: $ u_n$ is bounded in $ L^\infty(Q)$ ...
MATAKA's user avatar
  • 53
4 votes
1 answer
285 views

Derivatives of diffeomorphism whose iterates on an open set converge to a point

Consider a smooth manifold $M$, a diffeomorphism $\varphi\in\mathrm{Diff}^\infty(M)$, and an open subset $B\subseteq M$. Suppose that, when restricted to $B$, $\varphi^n$ converges uniformly to a ...
user815293's user avatar
20 votes
1 answer
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Does every series of hyperreal numbers converge to some hyperreal number?

I am currently trying to find some field $F$ which includes $\mathbb{R}$ (or $\mathbb{C}$) and in which series $x^* = \sum_{i\in\mathbb{N}} x_i$ converge to some element of the field. (i.e. $x^* \in ...
Gilbert Bernstein's user avatar
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An orthonormal sequence of functions with sufficient pointwise cancellation

I suspect that someone immediately knows an answer to this question. I am looking for an infinite sequence of real-valued continuous functions $f_j:[0,1]\rightarrow\mathbb{R}$ such that $f_1,f_2,\...
alext87's user avatar
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0 votes
0 answers
108 views

Sum with the fractional part function

Helo, I am still interested in the asymptotic behavior of certain sums of type $\sum_{k=1}^{n}\left\{ \frac{h(n)}{h(k)}\right\}$ and here I conjecture that we have $$\sum_{k=1}^{n}\left\{ \frac{2^{n}+...
 Babar's user avatar
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1 answer
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Rate of convergence of the minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
MathLearner's user avatar
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1 answer
76 views

Decay rate of minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
MathLearner's user avatar
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0 answers
99 views

Does the following series converge? To which value?

Given $f(x)=A \cdot x^5+B \cdot x^8$ with: $A \in \mathbb{R}^-$ $B \in \mathbb{R}^-$ $h(z) = w_0 + \sum_{n=1}^\infty h_n \cdot \frac{\left(z-f(w_0)\right)^n}{n!}$ $w_0=1$ and: $$h_n=\lim_{w \...
Arthur's user avatar
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1 vote
0 answers
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Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$

The following question was asked on Math Stack Exchange by me 15 days ago. I used a bounty, but still no response. So I am posting the question here. Here is the link of the question here. problem ...
Mods And Staff Are Not Fair's user avatar
6 votes
2 answers
436 views

Asymptotic behavior of the "Cauchy square" series

$\renewcommand{\ge}{\geqslant}\renewcommand{\le}{\leqslant}$ $\newcommand{\pa}[1]{\left( #1 \right)}$ Let us take $\alpha > 0$, $x_1 := \alpha$ and for any $n \ge \mathbb{N}$, \begin{align*} \boxed{...
Raphaël's user avatar
2 votes
1 answer
293 views

On properties of sums involving the floor function

During my research on properties of fractional part and integer part functions, I was led to consider the function of two variables $f(n,k)=\frac{2^{k}+1}{2^{ n}+1}\left\lfloor \frac{2^{n}+1}{2^{k}+1}\...
 Babar's user avatar
  • 405
9 votes
2 answers
630 views

Another limit involving the fractional part

It is known that $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left\{ \frac{n}{k}\right\} =1-\gamma$$ where $\left\{ x\right\}$ is the fractional part of $x$ and $\gamma$ is the Euler constant. ...
 Babar's user avatar
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5 votes
1 answer
170 views

Converging paths implies converging parallel transports along those paths?

Suppose we have a vector bundle $E$ with connection $\nabla$ over a smooth manifold $M$. Let’s also say we have a sequence of smooth paths $\gamma_n\in C^\infty([0,1],M)$ starting at the same point $\...
user815293's user avatar
2 votes
0 answers
73 views

Limit of lacunar power series at $1^-$

I've asked this question on MSE but I didn't get a convincive answer so I'm trying here. Here is the question : Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider ...
Tuvasbien's user avatar
  • 176
2 votes
1 answer
165 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
Kanydo Mat's user avatar
0 votes
0 answers
41 views

Formalization of sample convergence

Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that $$X_i = \sigma_i z_i,$$ where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\...
Grigori's user avatar
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0 votes
0 answers
74 views

Convergence of probabilities imply convergence of joint probability

Context: Suppose I have two pairs of sequences of random variables $X_n, \tilde{X}_n$ and $Y_n, \tilde{Y}_n$, where $X_n$ and $Y_n$ are not necessarily independent for any $n$, but $\tilde{X}_n$ and $\...
Grigori's user avatar
  • 33
2 votes
1 answer
325 views

Convergence of distance

Consider these sets $$ A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])=0\} $$ where: ...
Star's user avatar
  • 76
0 votes
0 answers
74 views

Convergence in probability of quadratic form with positive mean

Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
Student88's user avatar
  • 503
1 vote
1 answer
148 views

Bound on a two-dimensional recursive series

For $n,k\in\mathbb{N}$, let $f(n,k)$ be defined as follows. If $n \geq k$ and $n > 2$, then $$ f(n,k) = \frac{k(n-k)}{n(n-1)}f(n-2,k-1) + \frac{k(k-1)}{n(n-1)}f(n-2,k-2) + \frac{n-k}{n}f(n-1,k) + \...
macat's user avatar
  • 145
0 votes
1 answer
78 views

Uniform convergence of differential quotients in $L^1$

I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
Mario Vasilija's user avatar
2 votes
2 answers
116 views

"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$

Let $H$ be a separable Hilbert space with the inner product $\langle, \rangle$ and $\{ T_n \}$ be a sequence of unbounded closed linear operators with a common dense domain $D \subset H$ such that $...
Isaac's user avatar
  • 3,091
1 vote
1 answer
78 views

Mittag-Leffler expansions converging to bounded function

Is it true that $$\lim_{N\to\infty}\left\langle\sum_{n=-N^2}^{N^2}\frac1{(Nx-n)^2}\right\rangle_N=\pi^2$$ for some suitable definition of "minima smoothing" such as $\langle f(x)\rangle_N\...
Adam's user avatar
  • 113
0 votes
0 answers
43 views

How to find A(i, d)?

Let $s(n)$ denote the digit sum of a natural number $n$. For $i, d\in \mathbb{N}$ define $$A(i, d) = \limsup_{m\to \infty}\frac{|\{n\leq m | s(n)\equiv i\mod d\}|}{m}.$$ Compute $A(i, d)$ for all $i, ...
Drrd's user avatar
  • 11
1 vote
2 answers
491 views

Show convergence result

Consider the following sets: $$ A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\}, $$ and $$ A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...
Star's user avatar
  • 76
2 votes
1 answer
104 views

Lipschitz continuity of eigenprojections

This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
Guanaco96's user avatar
1 vote
1 answer
114 views

Limit of $F_{n}(\lfloor{nx}\rfloor)$ where $ F_{n}(k)=G_{n}(k)+H_{n}(k)F_{n}(k+1) $ and $F_{n}(n)=\mu.$

The following conjecture is inspired by asymptotic results in generalizations of the secretary problem. CONJECTURE Consider a sequence of functions {$F_n$} with $F_{n}:[0,n]\cap \mathbb{Z}\rightarrow\...
José María Grau Ribas's user avatar
0 votes
1 answer
372 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
S.H.W's user avatar
  • 61
7 votes
1 answer
541 views

A variation on the Borel–Cantelli lemma theme

Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let \begin{equation*} E:=\bigcap_{n\ge0}B_n, \end{equation*} where \begin{equation*} B_n:=\...
Iosif Pinelis's user avatar

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