Questions tagged [limits-and-convergence]

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

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Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes

Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...
Fam's user avatar
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Normal numbers and law of the iterated logarithm

If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
Vincent Granville's user avatar
3 votes
1 answer
140 views

Localic maps given by series

Maps between real numbers are often defined by convergent series. For example, to define the exponential map, we can just prove that series $$ \sum_{n = 0}^{\infty} \frac{x^n}{n!} $$ converges, which ...
Valery Isaev's user avatar
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Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality

It is known that the fractional $L^p$ Sobolev inequality $$ \|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy $$ ...
julian haddad's user avatar
17 votes
2 answers
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"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

Recently, I encountered this problem: "Given a sequence of positive number $(x_n)$ such that for all $n$, $$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$ Find the limit $\lim_{n \rightarrow \infty} \...
Paresseux Nguyen's user avatar
1 vote
0 answers
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If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?

Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
Vincent Granville's user avatar
1 vote
1 answer
176 views

Does pointwise convergence yield the convergence under Skorokhod topology?

Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
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Interchange summation order in the limit of number of elements going to $\infty$

Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...
user1172131's user avatar
2 votes
1 answer
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Law of large numbers for triangular arrays whose moments "look independent"

Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that $$ \mathbb{E}\left[\...
Greg Zitelli's user avatar
1 vote
0 answers
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"Beautiful convergence" in Hejhal, The Selberg Trace Formula

A simple question: Hejhal in his volumes on the Selberg Trace formula (particularly volume 2) uses the expression "converges beautifully". I can't find the definition, even searching the ...
Jonathan Weitsman's user avatar
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Dense subspace of square integrable functions on the complex disc

Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...
Assinisa Hamidata's user avatar
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Taming families of rate functions

$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$. Let us say that a family $(r_j)_{j\in J}$ of ...
Iosif Pinelis's user avatar
9 votes
2 answers
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Rearrangement, conditional convergence, and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
Noah Schweber's user avatar
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1 answer
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Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
user1172131's user avatar
3 votes
0 answers
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On tangential approach regions for general power series converging on the unit disk

Notation and premises. Here it is a list of notations more or less explicitly used in the question: If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(...
Daniele Tampieri's user avatar
2 votes
0 answers
86 views

Relations between different "propagation of chaos" type results?

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...
Fei Cao's user avatar
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-2 votes
1 answer
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Tricky (for me) limit

I've been trying to compute the following limit for a few hours. Let $f(\gamma, \beta)$ be defined as follows: $$f(\gamma, \beta)=\lim_{x \rightarrow \infty} (1-\gamma^{1/x})(\log(x))^{\beta}.$$ I am ...
colin's user avatar
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What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$? $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n $$ I know it is convergent at least ...
mttt's user avatar
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4 votes
1 answer
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Mikusiński's approach to Bochner integrals; replace absolute by unconditional?

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions: Defn. Let $X$ be a Banach space. ...
Willie Wong's user avatar
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1 vote
3 answers
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Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
Alex M.'s user avatar
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4 votes
1 answer
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Distance between trunctated random walk and its normal form

I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$ $$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...
nodis6's user avatar
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2 votes
1 answer
276 views

One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges $$ \begin{split} \sum_{1<n\leq N}\frac{a_{n}}{\...
tongyang2357's user avatar
3 votes
1 answer
216 views

For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?

Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...
cgmil's user avatar
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1 answer
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Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
Emilio Ferrucci's user avatar
-1 votes
1 answer
161 views

Convergence to a constant or not? Reference request [closed]

Consider the function $$f(n) = \log n /(n\ \log\theta(p_n)),$$ where $\theta$ is the first Chebyshev function and $p_n$ is the $n$-th prime. Does $f$ converge to a constant as $n$ grows to infinity, ...
EGME's user avatar
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0 votes
1 answer
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A problem of the limit of $\frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}}$

Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if $$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \...
Watheophy's user avatar
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Persistence of planar trajectory converging to a node / focus

I consider a planar system $\dot u =F(u,p)$ where p is a scalar parameter. Suppose that the flow $\phi^t(u_0; 0)$ from $u_0$ converges to a stable node / focus $x^{eq}_0$ for the parameter value $p=0$....
herve's user avatar
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3 votes
1 answer
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Boundedness and convergence

If I know that $\Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$ and that $\nabla \Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$, is it true that $\nabla \Phi_\varepsilon \...
Markus's user avatar
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0 answers
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$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $

$f(x)$ is continuous for $\forall x \geq 0$ and monotonically decrease. $f(0)=0$. $a>0$. Is it true $$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $...
Watheophy's user avatar
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1 vote
1 answer
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Compare two limits related to Hölder condition

Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that $$ \varlimsup_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\...
Watheophy's user avatar
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2 votes
2 answers
515 views

About roots of polynomials [closed]

Let $n\in\mathbb N^*$, $P(x)=a_0+\dotsb+a_{n-1}x^{n-1}+x^n$ and $r_1,\dotsc,r_n\in\mathbb C$ the roots of $P$. Is it true $\lim\limits_{\max(\lvert a_i\rvert,i=0\dotsc n-1)\rightarrow 0} \max(\lvert ...
Dattier's user avatar
  • 3,737
2 votes
1 answer
375 views

Lower bound and limit of a sum with binomial coefficients

Let $$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$ $$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$ $$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
macat's user avatar
  • 145
5 votes
4 answers
817 views

Limit of a sum with binomial coefficients

Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ $$C_k = \frac{\sum_{i=1}^k(...
macat's user avatar
  • 145
4 votes
2 answers
669 views

Convergence almost everywhere of characteristic functions

Let $(\Phi_n)_n$ be the characteristic functions of probability measures $(\mu_n)_n$ and let $\Phi$ be the characteristic function of a probability measure $\mu$. Do you know an example where $\Phi_n(...
Tiblodocus's user avatar
2 votes
1 answer
113 views

Convergence of the average weight of an infinite path through a weighted directed graph

Consider a directed graph $G = (V, E, w)$, where $V$ is the set of vertices, $E \subseteq V \times V$ is the set of directed edges (with self-loops allowed), and $w : E \to \mathbb{R}_+$ is a weight ...
David's user avatar
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4 votes
2 answers
277 views

$\lim\inf_{t\to \infty} f(x(t))=0\Longrightarrow \lim\inf_{t\to\infty} \|x(t)\|=0$

I am a PhD student and during my research I was presented to the claim that For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, ...
Quiet_waters's user avatar
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0 answers
58 views

Is $g = \sum_{n \in \mathbb{Z}} f(\cdot - n)$ continuous if $f$ is vanishing, continuous, and integrable?

Let $f \in \mathcal{C}_0(\mathbb{R}) \cap L^1(\mathbb{R})$ be a continuous and integrable function such that $f(x) \rightarrow 0$ when $|x|\rightarrow \infty$. The sequence of a functions $f_N = \sum_{...
Goulifet's user avatar
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2 votes
0 answers
186 views

Uniform limit of pointwise limits of continuous functions

Let $X$ be topological spaces, $Y$ a metric space and $(f_n)_{n\in\mathbb{N}}$ a sequence of functions, with $f_n:X\rightarrow Y$ pointwise limit of continuous functions for each $n\in\mathbb{N}$. ...
Lorenzo's user avatar
  • 2,134
0 votes
1 answer
298 views

Random variable is Big O in probability notation

Let $R_n$ be a random variable with values in $[0,1]$ and $nR_n$ converges to $\frac{1}{1+C} \chi_m^2$ in distribution for some constant $C>0$ and $m\in \mathbb{N}$. Is it possible to show that $(1-...
Hugo10T's user avatar
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1 vote
0 answers
300 views

Convergence in law and distribution theory

A standard result in probability theory asserts that a sequence of probablity measures $\mu_n$ on the Borel $\sigma$-algebra of $\bf R$ converges in law or weakly to a probability measure $\mu$ if and ...
coudy's user avatar
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8 votes
3 answers
563 views

Why is there an unexpected increase in the density of certain types of Goldbach primes?

Note: Posted in MO since it was unanswered in MSE. I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the ...
Nilotpal Kanti Sinha's user avatar
2 votes
1 answer
141 views

Convergence of measure of products of random unitaries

I'm trying to read Convergence conditions for random quantum circuits by Emerson, Livine, Llyod (https://doi.org/10.1103/PhysRevA.72.060302), arXiv version: (https://arxiv.org/abs/quant-ph/0503210) ...
nervxxx's user avatar
  • 207
17 votes
1 answer
890 views

Fraction of $S_n$ reachable by using every transposition once as $n\to\infty$?

For $n\in \mathbb{N}$ let $S_n$ denote the set of permutations (bijections) $\pi: \{0,\ldots,n-1\}\to \{0,\ldots,n-1\}$. A transposition swaps exactly $2$ elements and is often denoted by $(i \; k)$ ...
Dominic van der Zypen's user avatar
4 votes
1 answer
229 views

Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number

Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
Nilotpal Kanti Sinha's user avatar
4 votes
1 answer
191 views

Generalized limits in Boolean algebras

Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a ...
Damian Sobota's user avatar
2 votes
0 answers
120 views

Average length of consecutive integers which have an increasing number of divisors

Consider the nine consecutive natural numbers starting from $1584614377$. ...
Nilotpal Kanti Sinha's user avatar
6 votes
1 answer
454 views

A limit problem

Let $f$ be a bounded and continuous function, $0<a < 1$. $U(x,r)$ is the neighborhood of $x$ with diameter $r$. Can we prove the following equation of two limits $$ \lim_{r\rightarrow 0} \sup_{...
Watheophy's user avatar
  • 419
1 vote
1 answer
153 views

Non-Archimedean Lebesgue dominated convergence theorem

In this paper, the authors explain that the full generality of the Lebesgue dominated convergence theorem holds for functions on a compact zero-dimensional space $X$ taking values in a metrically ...
MCS's user avatar
  • 1,256
2 votes
1 answer
174 views

Pointwise almost sure convergence implies global convergence

Sorry in advance if this is not sufficiently research-level, it is really more of a reference request since the proof is not difficult. Let $\mathcal{Y}$ be a compact set, let $\{X_n\}$ denote a ...
Tom Solberg's user avatar
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1 vote
0 answers
95 views

Sum of reciprocals of maximal prime gaps and primes

Let $G_r =$ http://oeis.org/A005250, and $P_r =$ http://oeis.org/A002386. $\sum_{n=1}^{\infty}{\frac{1}{G_r}} = c_1$ $\sum_{n=1}^{\infty}{\frac{1}{P_r}} = c_2$ Do the constants c_1 and c_2 exist? The ...
John Nicholson's user avatar

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