Questions tagged [limits-and-convergence]

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

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77 views

Integrate Radon-Nikodým derivatives against Lebesgue measure

I am struggling for quite some time, because of a problem involving Radon-Nikodým derivatives. I will try to describe the main features and perhaps somebody has an idea how to solve it. I consider two ...
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92 views

A limit calculation

I wonder if the limit below $$\lim_{x\rightarrow +\infty} e^{-x}\sum_{j=0}^{\infty}\frac{x^{j+a}}{\Gamma(j+a+1)}$$ equals 1, for real constant $a>0$, and how shall we get this result?
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Convergence of Radon-Nikodým derivative

Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
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151 views

Show that $\sup_{||g||\leq \delta_n}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^n g(Z_i)\right|\rightarrow_{a.s.}0.$ when $\delta_n\rightarrow 0$?

UPDATE: The result below can be understood as an almost sure stochastic equicontinuity condition. I don't know of any result establishing primitives of almost sure stochastic equicontinuity. If you ...
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89 views

Natural candidates for sub-half-exponential which limit to half-exponential function from below

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However sub-half-exponentials (functions whose composition grows ...
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87 views

Natural candidates for super-half-exponential which limit to half-exponential function from above

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However super-half-exponentials (functions whose composition grows ...
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When does this limiting ratio give a real root $x$ to the equation of the form $\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0$?

By searching the Inverse Symbolic Calculator, we appear to be able to make the following conjecture about a real root to the equation: $$\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0 \tag{1}$$ Let the ...
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The limit of the operator norm in a Hilbert space

I am not familiar with functional analysis. Could you tell me please, how to prove the following statement (if it is true)? $$ \lim_\limits{M \to \infty} \|T_A - T_b \| = 0, $$ here operator norm ...
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51 views

The scalar convergence in $\mathcal{C}(X)$ is topologizable?

Let $(X,\|.\|)$ be a separable Banach space and $\mathcal{C}(X)$ be the collection of all nonempty, closed and convex subsets of $X$. For any $C$ in $\mathcal{C}(X)$ we set $$ s(x^*, C) := \sup_{x\in ...
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Convergence for a non-linear second order difference equation

In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation $$ u_0,u_1>0, \qquad \forall n\in \mathbb N, \; u_{n+2}=a\ln(1+u_n)+b\ln(1+u_{n+1}) $$ with ...
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55 views

Finding a variable P for which a sum converges [closed]

I need guidance in finding a variable P for which $ \sum _{n=4}^{\infty }\:\left(\frac{n\ln \left(n\right)-n}{\ln \left(n!\right)}\right)^p $ converges, or proof that there doesn't exist such P ...
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260 views

Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$

I was working with some Dirichlet series and I realized that I have never seen any general conditions under which \begin{equation} \sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
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57 views

Convergence as measure vs in $H^{-2}$

Let the domain be the two dimensional torus $\mathbb{T}$, and let $f_n $ be a sequence bounded in $H^1$, such that $\sup_n |f_n|\le 1$, and $f_n \to f$ weakly in $H^1$. Let $u_n = f_n \frac{\partial ...
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For which values of $ x$ we have $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges?

The copy of this question is posted here I have tried to to determine values of real $x$ for which $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges but I can't , Presumably the set of values $\{ ...
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1answer
95 views

Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
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56 views

Limit of the convolution of derivative of Gaussian heat kernel

I'm looking for the following limit: $$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...
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1answer
142 views

Spherical harmonics expansion

In the context of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{...
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81 views

Analyze a complicated double summation

Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$F(t,f(x)):=\dfrac{1}{t}\Big(\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\...
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1answer
65 views

The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
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Finding the upper bound and uniform convergence

For every positive integer $m$, let $x_0^{(m)}$ < $x_1^{(m)}$ < ... < $x_m^{(m)}$ be $(m+1)$ distinct points in $[0, \pi]$ and let $p_m$ $\in$ $P_{m+2}$ be the Hermite interpolant of the ...
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166 views

Properties of Dirichlet series

I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me. With $c\in [0,1]$, $$f(n) = \pm 1,...
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479 views

Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]

Edit: I got rid of my old definitions. Everything should be clear now Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...
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1answer
147 views

Convergence in $C_c$ but not in $C$

Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
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1answer
71 views

Convergence in LB-spaces

Edit: Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
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Characterization of uniformly convergent sequence of functions

Let $f_n$ a sequence of continuos function from a metric space $(X,d)$ itself. If $f_n$ converge uniformly to a function $f $ and $\varphi$ is real valued uniformly continuos function defined on $X$...
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1answer
40 views

Countable convergence-determining class for weak convergence of probability measures

Suppose that $E$ is a Polish space. Portmanteau theorem asserts that a sequence $(\mu_n)$ of Borel probability measures weakly converges to a Borel probability measure $\mu$ (shortly, $\mu_n\overset{...
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if $\max_{z \in K} |\zeta(z+it)-f(z)|<\epsilon.$ then is this $\lim_{t\to \infty} \inf \frac {|\zeta'(z+it)|}{|f'(z)|} $ a finit limit?

Universality theorem of Riemann zeta function states that :Let $K$ be a compact subset with connected complement lying in the strip $\{1/2 < \operatorname{Re}(z)<1\}$, and let $f : K \rightarrow ...
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1answer
72 views

Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
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2answers
192 views

$| f_n |^p - | f |^p - | f_n-f |^p$ converges in distribution sense if $f_k$ converges almost everywhere and weakly to $f$?

Let $1<p<\infty$ and $f_n$ be a sequence in $L^p(S^1)$ that converges weakly to some $f$. Here $S^1$ is the circle so we are dealing with periodic functions. Let us see if $| f_n |^p - | f |^p -...
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1answer
49 views

Connected graphs $G$ with $\delta(G) > 1$ and long minimum size roundtrips

Let $G = (V,E)$ be a finite, connected, simple, undirected graph. By a roundtrip of $G$ we mean a map $r:\{0,\ldots,n\} \to V$ for some $n\in\mathbb{N}$ with the following properties: $r$ is ...
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1answer
87 views

Convergence of estimator given by a fixed point

Let $X$ be a non-negative random variable with cdf $F$ and define $$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function. Let $s_0$ be the unique fixed point of $G$. Now let $X_1,\dots,X_t$ ...
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56 views

Convergence of Bayesian posterior

Let $\Delta [0,1]$ denote the set of all probability distributions on the unit interval. Let $\mu \in \Delta [0,1]$ denote an arbitrary prior. Importantly, $\mu$ does not necessarily admit a density ...
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If $u_n \to u$ in $H^1_0(\Omega)$, does $\chi_{\{u_n = 0\}} \to g$ for some $g$ in some space, for a subsequence?

Let $\Omega$ be a bounded and smooth domain. Suppose we have $u_n \to u$ in $H^1_0(\Omega)$. We know that for a subsequence, $\chi_{\{u_n = 0\}} \rightharpoonup f$ to some $f$, weak-* in $L^\infty(\...
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About generalized binomial theorem and Grünwald-Letnikov fractional derivative

I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
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2answers
466 views

Geometry of Level sets of elliptic polynomials in two real variables

Updated: A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
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Tightness of a uniformly bounded sequence of functions integrated with respect to a semimartingale

I am reading this paper by Jacod, Jakubowski and Mémin. In the proof of Theorem 1.3 the authors define, for each $n\geq1$ the function $\phi_n$ by $\quad\phi_n(s)=i+1-ns,\quad\text{if } \frac{i}{n}&...
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Is this integral finite and how does it decay to zero?

I would like to know if the following is convergent/finite (it represents a bound from a truncated Legendre series approximation) \begin{equation} \varepsilon_n \leq \int_{-1}^1 \left(\int_{n\gg 1}^\...
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1answer
90 views

Weak-* convergence in $L^\infty((0,T)\times\Omega)$ implies weak-* convergence in $L^\infty(\Omega)$ for a.e. $t \in (0,T)$?

Let $\Omega$ be a bounded and smooth domain. Suppose I have a sequence of non-negative functions $u_n \in L^\infty((0,1)\times \Omega) \cap L^\infty((0,1);L^\infty(\Omega))$ with $$0 \leq u_n \leq 1 \...
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A ballot-casting problem

For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ denote the collection of subsets of $X$ with cardinality $\kappa$. If $n$ is a positive integer, let $[n]:=\{1,\ldots,n\}$. Let $V$ and $K$ be ...
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2answers
560 views

What does the abbreviation “p.p.” mean in the context of convergence [closed]

What does the abbreviation "p.p." mean when referring to convergence? E.g. in the following paper by Harry Pollard THEOREM. If $f \in L^p$ for some $p$ in the range $\tfrac{4}{3} <p < \infty$,...
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$L^p$-convergence of truncated Legendre series approximation to inverse error function $\text{erfinv}$

Bounding the $L^p$-error for an $n$-th order Legendre series approximation I have function $f\,\colon (-1, 1) \to \mathbb{R}$ where $f \in L^q(-1, 1)$ for any $q \geq 1$. I approximate $f$ using a ...
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1answer
93 views

Matrix iteration for non-negative matrices. Does it converge to some eigenvector?

Let $A$ be a non-negative (entrywise) matrix such that $A(1,1)>0$. Set $u=(1,0,0,...,0)^T$. Is it always true that there exists a non-negative eigenvector $v$ of $A$ such that $\lim_{n\rightarrow\...
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Density of integers related to the size of its order of appearance in the Fibonacci sequence

Let $z(n)=\min\{k>0 : n\mid F_k\}$. This function is known as the Fibonacci entry point (for example). A result of Sallé gives the sharpest upper bound for $z(n)$, namely, $z(n)\leq 2n$, for all $n$...
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73 views

Reference for the positive probability of convergence to a stable point of a stochastic approximation algorithm

Consider a stochastic approximation process with $$x_{t+1} = x_t + \frac{1}{t} (g(x_t)+u_t)$$ where $(u_s)_s$ is a sequence of i.i.d. shocks. Assume $g$ is Lipschitz, $u_t$ has finite variance, and ...
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79 views

Intuition behind the local limit theorem in hyperbolic groups

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Denote by $X_n$ the induced random walk. Finally, let $p_n=\mu^{*n}(e)=P_e(X_n=e)$. The local limit ...
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1answer
291 views

About $a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}.$

Let $A,B,C > 0$. Put $a_1 = A$ and $a_2 = B$ and, for integer $n > 2$, $$a_n = \frac{a_{n-1}(a_{n-1} + C)}{a_{n-2}}$$ and $$ T = \lim_{k \to \infty} \frac{a_k}{ a_{k - 1}}.$$ Notice the limit ...
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1answer
61 views

Closed on generic set implies closed set whole set [closed]

Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
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1answer
186 views

Graphons and Graphs

The situation is as follows: assume we have a sequence of simple weighted graphs $(G_n)_{n\in\Bbb{N}}$. For the terminology that follows I refer to Limits of dense graph sequences by László Lovász and ...
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104 views

Convergence in unbounded domains

Lemma. Let $\mu$ be a measure in $\mathcal{M}(\Omega)$ and let $(v_{n})$ be a sequence of functions in $W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ converging to a function $v$ in the weak topology of ...
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87 views

Uniform convergence over compacts subsets implies existence of a uiform convergente subsequence?

Let $H$ the group of all homeomorphisms of a locally compact second countable and totally bounded metric space $X$ onto itself, under the compact-open topology ($X$ is totally bounded if every ...

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