# Questions tagged [limits-and-convergence]

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

808
questions

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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 ...

0
votes

0
answers

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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 $\...

1
vote

1
answer

50
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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\}_{...

1
vote

1
answer

52
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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 ...

9
votes

1
answer

486
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### 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 $...

0
votes

1
answer

92
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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 ...

0
votes

1
answer

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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 ...

6
votes

2
answers

317
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### 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 &...

0
votes

0
answers

63
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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.
...

2
votes

1
answer

87
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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 $...

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 ...

1
vote

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(...

0
votes

1
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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)\...

1
vote

1
answer

97
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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 ...

1
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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 ...

9
votes

1
answer

780
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### 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 $A_nB_nC_n$, ...

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 ...

0
votes

1
answer

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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 ...

3
votes

0
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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.$...

0
votes

0
answers

104
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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 ...

0
votes

0
answers

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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.

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)$ ...

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 ...

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 ...

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votes

0
answers

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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,\...

0
votes

0
answers

108
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### 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}+...

0
votes

1
answer

48
views

### 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 ...

0
votes

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 ...

0
votes

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 \...

1
vote

0
answers

178
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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 ...

6
votes

2
answers

436
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### 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{...

2
votes

1
answer

293
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### 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}\...

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. ...

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 $\...

2
votes

0
answers

73
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### 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 ...

2
votes

1
answer

165
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### 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, ...

0
votes

0
answers

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### 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 $\...

0
votes

0
answers

74
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### 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 $\...

2
votes

1
answer

325
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### 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:
...

0
votes

0
answers

74
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### 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 ...

1
vote

1
answer

148
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### 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) + \...

0
votes

1
answer

78
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### 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 ...

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 $...

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\...

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, ...

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)...

2
votes

1
answer

104
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### 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 ...

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\...

0
votes

1
answer

372
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### 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'...

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:=\...