# Questions tagged [limits-and-convergence]

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

363
questions

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votes

**1**answer

217 views

### Prove an existing formula for a limit of a specific sum

Prove that$$\lim_{n\to\infty}\frac1n\sum_{i_1,i_2,...i_k=1}^n\lambda_1^{|i_1-i_2-s_1|}\lambda_2^{|i_2-i_3-s_2|}...\lambda_k^{|i_k-i_1-s_k|}$$is equal to$$\sum_{j=1}^k\lambda_j^{S+k-1}\prod_{l=1,l\ne j}...

**1**

vote

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27 views

### Sufficient conditions for taking limits in stochastic partial differential problems

Let's say we have a (parabolic) Cauchy problem:
$$
(1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t))=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$
$$(2) \hspace{0.5cm} u(x,0)=u_0(x),
$$
...

**3**

votes

**1**answer

112 views

### What is the current fastest method to calculate Lerch's Phi transcendent?

Lerch's Phi transcendent is
$$
\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s}
$$
I am trying to compute this for the following parameters:
$z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (...

**0**

votes

**0**answers

72 views

### Is there any statistically convergent real sequence, which is not almost convergent?

I have read that almost convergence and statistical convergence are incompatible (i.e. not comparable).
For this both of below must be satisfied :
There exists a statistically convergent real ...

**1**

vote

**1**answer

85 views

### Limiting distribution of “scatter matrix” $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
...

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**0**answers

29 views

### Limits of the wave equation with piecewise constant propagation speed

This question is cross-posted from math.stackexchange.com, where it did not (yet?) get any answers despite a +100 bounty.
Consider a wave equation
$$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\...

**2**

votes

**0**answers

15 views

### Largest eigenvalue scaling in a certain Kac-Murdoch-Szegö matrix

I'm looking at $N\times N$ matrices $M_N$ with elements
$$M_N=\left( \rho^{|i-j|} \right)_{i,j=1}^N,$$
where $\rho$ is a complex number of unit modulus.
These matrices with $\rho\in\mathbb R$ and $|\...

**0**

votes

**0**answers

28 views

### Extension of a “statistical limit functional”

$c=$The set of all real convergent sequences
$l_\infty=$The set of all real bounded sequences
Clearly $c\subset l_\infty$
$f:c\to \mathbb R$ is called limit functional defined by $f(x)=\lim\limits_{...

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votes

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59 views

### Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function?
I tried to ...

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vote

**0**answers

34 views

### Convergence acceleration of a series by using optimal parameters

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...

**5**

votes

**1**answer

117 views

### Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...

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vote

**0**answers

29 views

### Linear programming with a convergent coefficient

The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...

**0**

votes

**2**answers

104 views

### Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$

We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim_{n\to\infty}y_{\alpha n}=0$) of rational numbers with the property: $\forall\...

**0**

votes

**1**answer

141 views

### Sum of reciprocal quadratic

Is there a general method or formula for calculating the infinite sum $\sum_{n=1}^{\infty} 1/(an^{2}+bn+c) $?

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65 views

### Does the intercept converge if we fit a best fit line to points with prime coordinates?

A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here.
Let $p_k$ denote the $k$th prime such that $...

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32 views

### Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...

**2**

votes

**1**answer

64 views

### Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$.
Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.
I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.
My first ...

**11**

votes

**1**answer

353 views

### Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...

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37 views

### If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^*$ = $\mathfrak{M}(G)$

Let $\mathfrak{M}(G)$ be the set of all means on $L_\infty (G)$
If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^* = \mathfrak{M}(G)$
My attempt:
We know ...

**1**

vote

**0**answers

51 views

### Limit Behavior of a Graph Iteration

,Let $G(V,E)$ be a weighted complete graph.
Let further $\min_k(v_i)$ denote, depending on whether the context is arithmetic or set theoretic, either the set of the $k$ smallest edges adjacent to $...

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votes

**1**answer

143 views

### Is the density of 1's in the Fibonacci word uniform?

The Fibonacci word is the limit of the sequence of words starting with $0$ and satisfying rules $0 \to 01, 1 \to 0$. Equivalently, it is obtained from the recursion $S_n= S_{n-1}S_{n-2}$ under ...

**1**

vote

**1**answer

139 views

### Why is this series summable?

Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$.
Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that
$\lim \sup_{k \rightarrow \infty} \frac{...

**3**

votes

**0**answers

53 views

### Does this definition of the Fourier intensity measure make sense?

Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$.
For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...

**7**

votes

**2**answers

236 views

### How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...

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vote

**2**answers

231 views

### Showing $o(1)$ convergence for ratio of successive binomial tail probabilities

For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...

**1**

vote

**0**answers

83 views

### Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$

Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$
Let $V= \...

**7**

votes

**0**answers

135 views

### A limiting sequence of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...

**5**

votes

**1**answer

134 views

### Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...

**10**

votes

**2**answers

382 views

### Value of $c$ such that $\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\log(k!)}=1$

What is the value of $c$ such that
$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\log(k!)}=1?$$
Numerically, it seems that the answer is $c=\log 2$. But I'd like to see a reason ...

**4**

votes

**0**answers

78 views

### Convergence acceleration of successions with logarithms

I have a numerical question regarding acceleration of a succession.
A preliminary: suppose that I have a succession $a_g$ that, for high $g$, asymptotically goes as
$$
a_g=s_0+\frac{s_1}g+\frac{s_2}{...

**7**

votes

**1**answer

155 views

### Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices
$$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$
Let
$$\pmatrix{\alpha_n & \beta_n \\...

**0**

votes

**1**answer

133 views

### What is the value of following limit?

Let $P$ be a polynomial in complex variable $z$ of degree $d$
i.e. $P(z)= a_d z^d+.....+a_1 z+a_0$
Now I want to calculate following limit
$f(z) = \limsup_{n \to \infty} \frac{1}{d^n} (Log|P(z)^{*...

**0**

votes

**2**answers

149 views

### Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...

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**0**answers

68 views

### Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

Setup
This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms.
So, let $Z$ be a $p$-dimensional random vector with (unknown) ...

**6**

votes

**1**answer

266 views

### Asymptotic behavior of a certain trigonometric partial sum

Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum:
$$
f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...

**-2**

votes

**2**answers

113 views

### What is the limit of this integral as $n$ approaches infinity for integer $k\geq 0$ and real $m\geq 1$? [closed]

$\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$

**1**

vote

**1**answer

149 views

### Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...

**4**

votes

**0**answers

264 views

### Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables:
$B^m_{1,1}$ $B^...

**6**

votes

**1**answer

274 views

### Weak*-convergence of signed measures

Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...

**0**

votes

**0**answers

143 views

### $L_1$ convergence for a product of indicator functions

Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions
$$
\lim_{N\...

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votes

**1**answer

281 views

### Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...

**1**

vote

**0**answers

45 views

### A convergence condition on tempered representation

Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the ...

**5**

votes

**0**answers

81 views

### Positive splitting of Sobolev convergence

Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...

**3**

votes

**0**answers

70 views

### Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...

**2**

votes

**0**answers

86 views

### Convergence Based on Recurrence Relation

I am studying a sequence based on the following recurrence:
$$X[t] = \sqrt{\alpha X[t-1]^2+(X[t-1]^2-\alpha X[t-2]^2)\frac{(2-X[t-1])^2}{X[t-1]^2}}$$
$$X[0]=0$$
$$X[1]>0$$
$$\alpha \in (0,1)$$
I ...

**2**

votes

**2**answers

75 views

### Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...

**3**

votes

**0**answers

60 views

### Limits of a simple damped system

Definition: Let $F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$ be the Laplace transform of $f_n(t)$.
Required Result: To show $\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$.
Ideas:
Let $G_n(s)=\frac{1}{s^{n+...

**1**

vote

**1**answer

209 views

### Intuition for coercive functions

I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?

**2**

votes

**3**answers

176 views

### Asymptotic forms of Legendre functions for large degree

Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are
$$
P_n(\cosh(x)) ~
\...

**3**

votes

**1**answer

328 views

### Expected value of the maximum of the periodogram

Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...