Questions tagged [limits-and-convergence]

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

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Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange; I am attempting to prove/disprove convergence of the following sum $$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p \left\{\frac{...
Brian's user avatar
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5 votes
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Rate of Convergence of Borwein Algorithm for computing Pi

In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. However there is a problem that I couldn't understand and couldn't ...
Sigma Park's user avatar
6 votes
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Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy ...
Cain's user avatar
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7 votes
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Factorial-based constant

Am looking for a name for: $$\prod\dfrac{1}{1-\dfrac{1}{n!}}$$ $$=2.529477472079152648180116154253954242$$ Wolfram|Alpha Expanding the formula gives: $$(1+\frac{1}{2!}+\frac{1}{2!^2}+\dots)(1+\...
JMP's user avatar
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8 votes
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Lemma in Scholze-Weinstein

In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7: Lemma: Let $K$ be a ...
Toby's user avatar
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Is $\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$ finite for every $k$?

I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$? where :$\phi_{k}$ is iterating Euler - totient function ...
zeraoulia rafik's user avatar
4 votes
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On contractive properties of a nonlinear matrix algorithm

I’m stuck in a problem that concerns a nonlinear iterative matrix algorithm. Although the problem is quite complicated to explain I’ll try to describe it in a simple way, neglecting unnecessary ...
Ludwig's user avatar
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0 votes
1 answer
339 views

Almost sure convergence of smallest eigenvector of diagonal matrix

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily ...
PThomasCS's user avatar
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convergence of unconstrained convex optimization

I encounter an optimization problem. The simplified version is like following: Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. $...
user5694458's user avatar
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1 answer
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Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D

Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) ...
Friedrich Philipp's user avatar
2 votes
0 answers
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Proving convergence is impossible for a sum of hyperbolic cosines

Suppose that $z$ is some complex value. Is it possible to prove that $$\lim_{n \rightarrow \infty} \sum_{j = 1}^n {\sqrt{n \over j}} \cdot \cosh(z \log {n \over j}-\operatorname{ Arccoth} (2z)) $$ ...
Nathan McKenzie's user avatar
2 votes
1 answer
411 views

Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $...
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4 votes
2 answers
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Uniform Convergence of Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
Rodrigo Zepeda's user avatar
6 votes
2 answers
383 views

Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
Agno's user avatar
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3 votes
2 answers
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Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows: $$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$ Upper bounds for $\phi$ can be simply derived from ending the product early, e.g. $$\phi(x)<\prod_{i=1}^...
R B's user avatar
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5 votes
1 answer
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The limit of the following product? What is the closed form of the value?

Assume that $P_n$ is the $n$'th prime: Please help me solve the following $$\lim_{k\to\infty} {k}\prod_{n=1}^k \frac{P_{2n-1}}{P_{2n}}$$ I am not really sure quite where to start here as I am ...
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3 votes
1 answer
364 views

Uniform convergence of 2-norm of a multinomial vector

Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e. $$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} p_1^{n_1}...
Hedonist's user avatar
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3 votes
0 answers
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Notion of convergence on a dense subset

My motivation for this question is as follows. Consider the set $D$ of cadlag functions on $(0,1)$ and its subset $D^\uparrow$ of non-decreasing cadlag functions. Each $f \in D$ has at most countably ...
yada's user avatar
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1 vote
1 answer
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limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit: $\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$ When $...
Bruno Brogni Uggioni's user avatar
5 votes
1 answer
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Weak convergence of random variables in $L^2$ and vague convergence

Dumb question: Let $X_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$. Suppose also that $\mu_n$, the distributions of $...
arjun's user avatar
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6 votes
1 answer
351 views

Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path. If we equip the space of continuous function starting from $0$, ...
yilin wang's user avatar
3 votes
1 answer
310 views

Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part. Let $a < b$ be two numbers in $[0, 1]$. Then by ...
Mohammad Ali Karami's user avatar
2 votes
2 answers
310 views

A term for sequences whose mean is defined?

This may be an extremely stupid and elementary question, but is there a name for sequences $\{a_i\}$ such that $\lim_{n\to\infty} \left( \frac{1}{n}\sum_{i=1}^{n} a_i\right)$ exists? This seems to be ...
Tom Solberg's user avatar
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3 votes
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Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$...
Qwertuy's user avatar
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2 answers
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Special filters in the algebra of regular open sets of a topological space

Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is: $$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)...
Rafał Gruszczyński's user avatar
2 votes
1 answer
478 views

Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?

Let $\{X_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n f(X_k)=\mathbb{E}(f(X_1))\ a.s.$$...
Samrat Mukhopadhyay's user avatar
2 votes
0 answers
348 views

Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
mic's user avatar
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45 votes
5 answers
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question. For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...
Michael Hardy's user avatar
1 vote
1 answer
2k views

Limit of largest eigenvalue [closed]

For positive definite matrix, if we increase the dimension to the infinity, is it true that the largest eigenvalue stays bounded from above? In other words does the following limit exists: $$\lim_{n\...
Tomas's user avatar
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4 votes
0 answers
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Convergence of a particular double sum [closed]

Consider the following double sum: $$Q(n)=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\left [ \partial _{ij}lnf\left ( x \right ) \right ]^2$$ where $\partial_{ij}$ is the partial second order ...
Tomas's user avatar
  • 267
-1 votes
1 answer
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Are limits decidable? Should definitions be decidable? [closed]

This question is about the Turing computability of the $\epsilon-N$ definition of a limit of an infinite sequence $S$. First, a proposition: There cannot exist a Turing Machine $M$ which, given a ...
Daniel Mansfield's user avatar
30 votes
1 answer
1k views

Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...
Michael Hardy's user avatar
2 votes
0 answers
99 views

Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$ does $$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$ hold for $$k(p,q)=\sum_{r=0}^{\infty}\sum_{l=...
warsaga's user avatar
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13 votes
2 answers
1k views

Is there a $q$-L'Hospital's Rule?

Let $\binom{n}{j}_q$ be a $q$-binomial coefficient and $(x;q)_n = (1-x)(1-qx)\cdots(1-q^{n-1}x).$ Consider the sum $$f(n,m,r,k)= \sum\limits_{j = 0}^{2n} {( - 1)}^{ j}q^{mj^2+rj} \binom{2n}{j}_{q^k}$...
Johann Cigler's user avatar
2 votes
0 answers
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Decomposition of the space according to the Ergodic Theorem

Given a space $(X, T)$, it is well known that for every $T$-invariant ergodic measure $\mu$, there exists a set $E_\mu$ of $\mu$-measure $1$ s.t. for every "nice" function $f$ $$ \frac{1}{n}\sum_{i=0}...
user18337's user avatar
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0 votes
0 answers
156 views

Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$: $h(\eta_{n}) \rightarrow \log2$...
Bruno Brogni Uggioni's user avatar
7 votes
3 answers
595 views

Natural topologies for the space of rational functions

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and,...
Arnold Neumaier's user avatar
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0 answers
105 views

Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)

Consider the following Volterra integral equation $g(t)=∫_0^tK_n(t,s)w_n(s)ds$ where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...
Kass's user avatar
  • 243
2 votes
0 answers
167 views

Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that $\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$ (stated, but not proved in "On ...
J Fabian Meier's user avatar
2 votes
1 answer
280 views

Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum: $$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$ when random variables $X_i$ ar i.i.d. Are there any investigation ...
Tomas's user avatar
  • 267
2 votes
1 answer
421 views

Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?

Let $G$ be a Lipshitz domain and $u_n \to u$ in $W_p^1(G)$. Is it correct, that $\frac{\partial |u_n|}{\partial x_m} \to \frac{\partial |u|}{\partial x_m}$ in $L_p(G)$? I know, that $\frac{\partial |...
Тимофей Ломоносов's user avatar
6 votes
2 answers
410 views

Convergence of Fixed-Point Iteration of a dependent map

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore ...
Rufio's user avatar
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1 vote
0 answers
150 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
pre-kidney's user avatar
  • 1,289
4 votes
1 answer
221 views

Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field. Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials $P_n:...
User28341's user avatar
  • 599
0 votes
1 answer
117 views

Convergence of generalized expectation under total variation norm

Consider the space of sub-distributions (i.e. positive measures of variation norm lower than 1) over a discrete subset $S$ of $\mathbb{R}$ (this set is measured by its powerset). Let $(\mu_n)_n$ be a ...
Justin Hsu's user avatar
0 votes
0 answers
53 views

On sequences of rational functions [duplicate]

Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \...
Pablo's user avatar
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3 votes
2 answers
906 views

Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent ("weakly")?

It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I ...
user228494's user avatar
2 votes
0 answers
322 views

Question about the characteristics of semimartingales

Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in D$....
CodeGolf's user avatar
  • 1,837
5 votes
1 answer
207 views

Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible. Let $a_k >0$ be an increasing sequence ...
RealMax's user avatar
  • 53
2 votes
1 answer
135 views

Variant of Skorokhod's theorem

Consider the following situation: $S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful). There is a a random variable $\zeta: \Omega \to S$. $f_n(\zeta) \to^d \eta$, i....
Nate Ackerman's user avatar