Questions tagged [limits-and-convergence]
Convergence of series, sequences and functions and different modes of convergence.
778
questions
4
votes
1
answer
415
views
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&...
7
votes
1
answer
1k
views
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}...
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 ...
4
votes
1
answer
231
views
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 ...
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 ...
0
votes
1
answer
68
views
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 ...
0
votes
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 $\...
3
votes
1
answer
651
views
$\{(\log n)^\alpha\}$ 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=...
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 ...
3
votes
1
answer
287
views
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 ...
0
votes
2
answers
176
views
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 ...
1
vote
0
answers
126
views
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 ...
1
vote
1
answer
176
views
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 ...
2
votes
1
answer
202
views
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 $\...
2
votes
1
answer
92
views
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 $...
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 ...
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, ...
13
votes
1
answer
3k
views
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,\...
2
votes
1
answer
112
views
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.
...
0
votes
0
answers
67
views
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 ...
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 ...
2
votes
1
answer
200
views
Limiting behavior of lattice sums
I suspect that
$$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \frac{1}{\sqrt{i^2+j^2}} =a\approx 1.76$$
$$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \frac{1}{\sqrt{...
4
votes
1
answer
269
views
How bad can pointwise convergence in $C$ be?
$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let
\begin{equation}
f(u):=\frac{2u^2}{1+u^4},
\end{equation}
so that the function $f\colon\R\to\R$ is continuous, $0\...
6
votes
1
answer
880
views
Convergence of Fourier series
Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions?
Say now $p = 1$, ...
2
votes
0
answers
124
views
A question about sequences of bounded variation and series convergence
There is a conclusion:
For any $x\in \mathbb R^\mathbb N$, we denote by $A_x$ the set $$A_x= \{a\in \mathbb R^\mathbb N:\sum_n x(n)\alpha(n)~\text{converges}\},$$
then for $y,x_1,x_2,\dots,x_k \in \...
2
votes
1
answer
103
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
3
votes
1
answer
144
views
About the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$
Let the sequence:
$s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$, for $mn>0$.
Computationally it seems that $\frac{s_{n+1}}{s_{n}} \approx e\cdot ...
1
vote
0
answers
180
views
Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution
Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
10
votes
1
answer
846
views
Pointwise convergence imples uniform convergence in an infinite subset
I came upon this statement in a stack answer.
Statement :
If $f_n$ is a sequence of real valued functions (not necessarily continuous or measurable) on $[0,1]$ such that $f_n$ converges point-wise to $...
4
votes
1
answer
132
views
CLT for a converging array of random variables
Assuming that for each fixed $k$, $(X_{n,1},\ldots,X_{n,k})\Longrightarrow(X_1,\ldots,X_k)$ where $X_1,\ldots,X_k$ are i.i.d. with mean zero and variance $\sigma^2$, will the array inherit the CLT ...
1
vote
1
answer
96
views
Convergence of discretized process when its predictable part converges to infinite variation process
This question seems to be related to Theorem IX.7.28 in J. Jacod and A. Shiryaev's Limit theorems for stochastic processes (2013), and it is very important to prove asymptotic properties of my ...
1
vote
1
answer
237
views
How to prove the convergence of this kind of sequence?
$$
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
$$
In fact, the specific problem (comes from probability theory) I want to solve is that:
...
8
votes
0
answers
284
views
On the remainder of a power series evaluated on the boundary of its convergence disk
Background
This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...
4
votes
0
answers
163
views
Sequence of minimal surfaces with bounded second fundamental form and area
Let $M^3$ be a closed orientable smooth manifold, let $g_n$ be a sequence of Riemannian metrics on $M$ converging to $g$ and let $\Sigma_n$ be a sequence of closed orientable $g_n$-minimal surfaces ...
3
votes
1
answer
156
views
Convergence of the solutions of a ODE system
Consider this system of differential equations for $t\in[0,\infty)$:
$$ \frac{d}{dt}x(t) = a(t) + F(x(t), y(t)),$$
$$ \frac{d}{dt}y(t) = a(t) + G(x(t), y(t)),$$
with positive initial conditions: $y(0)&...
1
vote
0
answers
95
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$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]
For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is
$$
\lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
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 $x_n$ ...
0
votes
0
answers
184
views
$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ 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 ...
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 $$a+b "=...
7
votes
1
answer
245
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
7
votes
2
answers
1k
views
What is Young measure?
I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem:
Theorem. Assume ...
3
votes
1
answer
256
views
Is there an asymptotic bound between converging and diverging series? [closed]
Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$,
$$
\log^{[k]}(x) =
\begin{cases}
\log^{[k-1]}(\log(x)) & k>0 \\
x & k=0
\end{cases}.
$$
It is well known, ...
0
votes
0
answers
96
views
Banach fixed point theorem / convergence squeeze
I am trying to prove a convergence result on an iterative scheme which has the initial point defined as
$$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$
where s(x) is some unknown function.
Here is my theorem and ...
2
votes
0
answers
75
views
Convergence of random operators
I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
4
votes
0
answers
221
views
Inductive limit of inclusions
Let $(\Lambda, \le)$ be a directed system and $\{ X_{\alpha} \}_{\alpha \in \Lambda}$ be a family of topological spaces indexed by $\Lambda$ such that $X_{\alpha} \subseteq X_{\beta}$ whenever $\alpha ...
1
vote
1
answer
242
views
Limit points and Homeomorphism
I was asking this question at Mathematics SE but I got nothing at all. This is why I am trying this site.
We consider the topology of the extended real line. Let $h\colon [-\infty,\infty]\to\Bbb R$ ...
1
vote
0
answers
62
views
The rate of convergence of Markov chain to stationary distribution
Let $X_t$ is Markov chain with transition rates $c: G \times G \rightarrow [0: +\infty)$, where $c(x, y) > 0$, $c(x, x) = -\sum_y c(x, y)$ for $x \neq y$. If $\mu_t(x)$ is the distribution of chain ...
3
votes
1
answer
395
views
Functions with at most linear growth at infinity: is the constant itself continuous?
I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M_f$ such that
\begin{...
4
votes
0
answers
290
views
Relationship between Hausdorff convergence of sets and indicator functions
Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.:
$$
\lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\...