Questions tagged [limits-and-convergence]
Convergence of series, sequences and functions and different modes of convergence.
778
questions
2
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1
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113
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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 $(X,Y)$ ...
1
vote
0
answers
150
views
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 ...
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 ...
5
votes
0
answers
164
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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
$$
\|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy
$$
...
17
votes
2
answers
2k
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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} \...
1
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0
answers
111
views
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 $L(s,\chi)$...
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. $\lim_{n\to\infty }f_n(t)$ ...
1
vote
1
answer
2k
views
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 ...
2
votes
1
answer
291
views
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[\...
1
vote
0
answers
129
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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 ...
0
votes
0
answers
65
views
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-...
0
votes
0
answers
46
views
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 ...
9
votes
2
answers
377
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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 ...
0
votes
1
answer
340
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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 ...
3
votes
0
answers
117
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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(...
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 ...
-2
votes
1
answer
201
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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}.$$
I am ...
0
votes
0
answers
127
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What's the convergence condition for the generating function formula of Legendre polynomials?
What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$?
$$
\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n
$$
I know it is convergent at least ...
4
votes
1
answer
261
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Mikusiński's approach to Bochner integrals; replace absolute by unconditional?
In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions:
Defn. Let $X$ be a Banach space. ...
1
vote
3
answers
486
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Squeezing more convergence from the convergence in all $L^p$ spaces
Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
4
votes
1
answer
85
views
Distance between trunctated random walk and its normal form
I have $$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) \...
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
$$
\begin{split}
\sum_{1<n\leq N}\frac{a_{n}}{\...
3
votes
1
answer
216
views
For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?
Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...
2
votes
1
answer
217
views
Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?
It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
-1
votes
1
answer
161
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Convergence to a constant or not? Reference request [closed]
Consider the function
$$f(n) = \log n /(n\ \log\theta(p_n)),$$
where $\theta$ is the first Chebyshev function and $p_n$ is the $n$-th prime. Does $f$ converge to a constant as $n$ grows to infinity, ...
0
votes
1
answer
106
views
A problem of the limit of $\frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}}$
Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if
$$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \...
0
votes
0
answers
43
views
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 $p=0$....
3
votes
1
answer
91
views
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 \...
1
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0
answers
77
views
$ \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
$$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $...
1
vote
1
answer
53
views
Compare two limits related to Hölder condition
Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that
$$ \varlimsup_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\...
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 ...
2
votes
1
answer
375
views
Lower bound and limit of a sum with binomial coefficients
Let
$$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$
$$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$
$$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
5
votes
4
answers
817
views
Limit of a sum with binomial coefficients
Let $$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(...
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(...
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 ...
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 $x$, ...
0
votes
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_{...
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 $n\in\mathbb{N}$. ...
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-...
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 ...
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 ...
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) ...
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 $(i \; k)$ ...
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 ...
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 ...
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 $1584614377$.
...
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_{...
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 ...
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 ...
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 ...