Questions tagged [limits-and-convergence]

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

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Speed of convergence of $\zeta(2k)\to 1$?

From the definition of $\zeta(z):= \sum_{k=1}^\infty \tfrac{1}{k^z}$ for $\mathrm{Re}(z)>1$ it is obvious that $\zeta(2k)\downarrow 1$ as $k \rightarrow \infty$. I am interested in the "true&...
Iceman's user avatar
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Summary of sufficient conditions for convergence of Fourier series

I would like to summarize various sufficient conditions for various modes of convergence of Fourier series. The followings are what I have gathered so far: $L^p$ convergence: if $f \in L^p(\mathbb{T}...
user141240's user avatar
1 vote
0 answers
82 views

Bound for specific series of convergent series

For $n \in \mathbb{N}$ and $k \in \{1,\ldots,n-1\}$ we define $$ a_n^k := n\left( \frac{(n-1)\cdots(n-k)}{n^k} - 1 \right) \ . $$ Using l'Hospitals rule iteratively for a total of $k-1$ times one can ...
Tardis's user avatar
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4 votes
1 answer
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Finding closed forms/related constants to a limit involving tetration

I was working on finding a series expression for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x^y) = f(x)^{f(y)}$ along the way for construction of such a function I came across a ...
Sidharth Ghoshal's user avatar
2 votes
1 answer
131 views

How to compute this limit involving the associated Legendre function?

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...
Student's user avatar
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Evaluating a limit at a discontinuity of a monotone rearrangment (distribution function)

I have a question that occurred to me and has been bothering me, because maybe graphically it seems obvious but I don't know how to get there. It has to do with the distribution function and monotone ...
NoetherNerd's user avatar
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1 answer
310 views

Almost sure convergence of the supremum over a class of random variables

Let $\mathcal{X}_n=\{ X_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\...
Jack London's user avatar
3 votes
1 answer
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$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?

The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
Vincent Granville's user avatar
1 vote
1 answer
819 views

Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does weak-* convergence for a sequence of functions $\{f_k\}_{k \in \mathbb{N}}$ in $W^{1,\infty}(\Omega)$ mean? It seems to me that there ...
vampip's user avatar
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1 answer
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Filling cups and buckets continuously

There are $n$ cups labeled $1,\dots,n$, each with a water tap that adds water into it at the same rate. There are also $k$ buckets, and $k$ sets $S_1,\dots,S_k\subseteq\{1,\dots,n\}$. At any point, if ...
pi66's user avatar
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What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...
dohmatob's user avatar
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1 vote
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Numerical calculation of a double integral from the slowly-decaying oscillating function

Let us consider the following integral $$ I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right]. $$ We know several properties of these functions. There are ...
MightyPower's user avatar
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1 answer
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Relation between two notions intermediate between “pointwise convergence” and “uniform convergence”

(I asked this on MSE a week ago, but did not get any answers there, so I'm trying here.) Let $X$ be a topological space. I will define four ways in which a sequence $(f_n)$ of continuous functions $X ...
Gro-Tsen's user avatar
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2 votes
1 answer
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Baire 1 function equivalence in measure

I am trying to prove (or disprove) the following assertion: Consider a probability triple $(X,\mathcal{B},\mu)$, $X$ separable Banach space (complete), $\mathcal{B}$ the Borel $\sigma-$algebra and $\...
Gioppa's user avatar
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2 votes
1 answer
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Convergence of sequences for Baire-1 functions

Let X and Y be separable Banach spaces. Let $f:X\rightarrow Y$ be a Baire-1 function, which is the pointwise limit of a sequence of continuous functions $f_n:X\rightarrow Y$. Define $E$ as the set of $...
Gioppa's user avatar
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6 votes
2 answers
279 views

Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?

Consider a continuous ODE, $$\dot x = f(x), f \in C^1$$ $\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
Concu Bine's user avatar
2 votes
0 answers
55 views

Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?

The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete? Indexing any countable set, ...
saolof's user avatar
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13 votes
1 answer
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Behavior of $n^\alpha \sin^{\circ\, n}(n^{-\alpha}x)$

I'll write it formally: Let $\sin^{\circ\, 1}(x) = \sin(x)$ and $\sin^{\circ n+1}(x) = \sin\bigl(\sin^{\circ n}(x)\bigr)$ for $n\in \Bbb N$ with $n>1$. What is the limit as $n \to \infty$? It's ...
18 votes
1 answer
691 views

Is the p-adic density of the image of a polynomial always rational?

This question was previously posted here on MSE. Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. For $n\in\mathbb N$, let $I_n$ be the number of integers $i\in\{1,\...
Riemann's user avatar
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2 votes
1 answer
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A question on the applicability Chebyshev inequality for sequence of random quantities

Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function. ...
dohmatob's user avatar
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LLN of random nearest neighbor function

There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
qwert's user avatar
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2 votes
2 answers
1k views

Convergence in probability of series of random variables

From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it ...
AlbertRapp's user avatar
2 votes
1 answer
200 views

Limiting behavior of lattice sums

I suspect that $$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \frac{1}{\sqrt{i^2+j^2}} =a\approx 1.76$$ $$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \frac{1}{\sqrt{...
Matt Majic's user avatar
4 votes
1 answer
269 views

How bad can pointwise convergence in $C$ be?

$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let \begin{equation} f(u):=\frac{2u^2}{1+u^4}, \end{equation} so that the function $f\colon\R\to\R$ is continuous, $0\...
Iosif Pinelis's user avatar
6 votes
1 answer
880 views

Convergence of Fourier series

Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions? Say now $p = 1$, ...
AleNekro97's user avatar
2 votes
0 answers
124 views

A question about sequences of bounded variation and series convergence

There is a conclusion: For any $x\in \mathbb R^\mathbb N$, we denote by $A_x$ the set $$A_x= \{a\in \mathbb R^\mathbb N:\sum_n x(n)\alpha(n)~\text{converges}\},$$ then for $y,x_1,x_2,\dots,x_k \in \...
Bo Peng's user avatar
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2 votes
1 answer
103 views

Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables

This is related to one of my previous questions here. Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
De vinci's user avatar
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3 votes
1 answer
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About the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$

Let the sequence: $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$, for $mn>0$. Computationally it seems that $\frac{s_{n+1}}{s_{n}} \approx e\cdot ...
José María Grau Ribas's user avatar
1 vote
0 answers
180 views

Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution

Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
moe.dancer's user avatar
10 votes
1 answer
846 views

Pointwise convergence imples uniform convergence in an infinite subset

I came upon this statement in a stack answer. Statement : If $f_n$ is a sequence of real valued functions (not necessarily continuous or measurable) on $[0,1]$ such that $f_n$ converges point-wise to $...
Kr Dpk's user avatar
  • 203
4 votes
1 answer
132 views

CLT for a converging array of random variables

Assuming that for each fixed $k$, $(X_{n,1},\ldots,X_{n,k})\Longrightarrow(X_1,\ldots,X_k)$ where $X_1,\ldots,X_k$ are i.i.d. with mean zero and variance $\sigma^2$, will the array inherit the CLT ...
mdou's user avatar
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1 vote
1 answer
96 views

Convergence of discretized process when its predictable part converges to infinite variation process

This question seems to be related to Theorem IX.7.28 in J. Jacod and A. Shiryaev's Limit theorems for stochastic processes (2013), and it is very important to prove asymptotic properties of my ...
Seung Hyeon Yu's user avatar
1 vote
1 answer
237 views

How to prove the convergence of this kind of sequence?

$$ x_{n}=\sum^{n-1}_{i=0} {a_i x_{n-1-i}} $$ where $$ \sum^{+\infty}_{i=0} {a_i}=1,1>a_i>0,1>x_i>0 $$ In fact, the specific problem (comes from probability theory) I want to solve is that: ...
zhjzwlys's user avatar
8 votes
0 answers
284 views

On the remainder of a power series evaluated on the boundary of its convergence disk

Background This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...
Daniele Tampieri's user avatar
4 votes
0 answers
163 views

Sequence of minimal surfaces with bounded second fundamental form and area

Let $M^3$ be a closed orientable smooth manifold, let $g_n$ be a sequence of Riemannian metrics on $M$ converging to $g$ and let $\Sigma_n$ be a sequence of closed orientable $g_n$-minimal surfaces ...
Eduardo Longa's user avatar
3 votes
1 answer
156 views

Convergence of the solutions of a ODE system

Consider this system of differential equations for $t\in[0,\infty)$: $$ \frac{d}{dt}x(t) = a(t) + F(x(t), y(t)),$$ $$ \frac{d}{dt}y(t) = a(t) + G(x(t), y(t)),$$ with positive initial conditions: $y(0)&...
moonlight's user avatar
1 vote
0 answers
95 views

$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]

For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is $$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
Ad_M's user avatar
  • 11
2 votes
1 answer
75 views

Convergence of localic maps

We can define a limit of a sequence of points in a locale in the usual way: $x$ is a limit of $\{ x_i \}_{i \in \mathbb{N}}$ if, for every open $U$ containing $x$, there exists $N$ such that $x_n$ ...
Valery Isaev's user avatar
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0 votes
0 answers
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$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?

For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold? $$ \left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right) $$ I ...
John's user avatar
  • 195
5 votes
1 answer
239 views

Hamming distance between $a+b$ and $a \oplus b \oplus ((a \land b) \ll 1)$

Motivation. In their paper about the cryptographic scheme NORX, the authors use a fast approximation of + by bitwise operations (taking fewer CPU cycles than proper addition) using the formula $$a+b "=...
Dominic van der Zypen's user avatar
7 votes
1 answer
245 views

Normal distribution by successive approximation?

$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
Iosif Pinelis's user avatar
7 votes
2 answers
1k views

What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
Devashish Sonowal's user avatar
3 votes
1 answer
256 views

Is there an asymptotic bound between converging and diverging series? [closed]

Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$, $$ \log^{[k]}(x) = \begin{cases} \log^{[k-1]}(\log(x)) & k>0 \\ x & k=0 \end{cases}. $$ It is well known, ...
Niv Sarig's user avatar
0 votes
0 answers
96 views

Banach fixed point theorem / convergence squeeze

I am trying to prove a convergence result on an iterative scheme which has the initial point defined as $$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$ where s(x) is some unknown function. Here is my theorem and ...
Doc Stories's user avatar
2 votes
0 answers
75 views

Convergence of random operators

I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
metric's user avatar
  • 121
4 votes
0 answers
221 views

Inductive limit of inclusions

Let $(\Lambda, \le)$ be a directed system and $\{ X_{\alpha} \}_{\alpha \in \Lambda}$ be a family of topological spaces indexed by $\Lambda$ such that $X_{\alpha} \subseteq X_{\beta}$ whenever $\alpha ...
genfuntranslate's user avatar
1 vote
1 answer
242 views

Limit points and Homeomorphism

I was asking this question at Mathematics SE but I got nothing at all. This is why I am trying this site. We consider the topology of the extended real line. Let $h\colon [-\infty,\infty]\to\Bbb R$ ...
00GB's user avatar
  • 179
1 vote
0 answers
62 views

The rate of convergence of Markov chain to stationary distribution

Let $X_t$ is Markov chain with transition rates $c: G \times G \rightarrow [0: +\infty)$, where $c(x, y) > 0$, $c(x, x) = -\sum_y c(x, y)$ for $x \neq y$. If $\mu_t(x)$ is the distribution of chain ...
Max Babich's user avatar
3 votes
1 answer
395 views

Functions with at most linear growth at infinity: is the constant itself continuous?

I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M_f$ such that \begin{...
A. Pesare's user avatar
  • 182
4 votes
0 answers
290 views

Relationship between Hausdorff convergence of sets and indicator functions

Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.: $$ \lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\...
SetValued_Michael's user avatar

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