Questions tagged [limits-and-convergence]

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

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58 views

LLN of random nearest neighbor function

There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
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2answers
119 views

Convergence in probability of series of random variables

From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it ...
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1answer
186 views

Limiting behavior of lattice sums

I suspect that $$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \frac{1}{\sqrt{i^2+j^2}} =a\approx 1.76$$ $$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \frac{1}{\sqrt{...
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1answer
215 views

How bad can pointwise convergence in $C$ be?

$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let \begin{equation} f(u):=\frac{2u^2}{1+u^4}, \end{equation} so that the function $f\colon\R\to\R$ is continuous, $0\...
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1answer
492 views

Convergence of Fourier series

Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions? Say now $p = 1$, ...
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0answers
96 views

A question about sequences of bounded variation and series convergence

There is a conclusion: For any $x\in \mathbb R^\mathbb N$, we denote by $A_x$ the set $$A_x= \{a\in \mathbb R^\mathbb N:\sum_n x(n)\alpha(n)~\text{converges}\},$$ then for $y,x_1,x_2,\dots,x_k \in \...
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1answer
49 views

Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables

This is related to one of my previous questions here. Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
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1answer
133 views

About the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$

Let the sequence: $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$, for $mn>0$. Computationally it seems that $\frac{s_{n+1}}{s_{n}} \approx e\cdot ...
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74 views

Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution

Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
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1answer
302 views

Pointwise convergence imples uniform convergence in an infinite subset

I came upon this statement in a stack answer. Statement : If $f_n$ is a sequence of real valued functions (not necessarily continuous or measurable) on $[0,1]$ such that $f_n$ converges point-wise to $...
4
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1answer
96 views

CLT for a converging array of random variables

Assuming that for each fixed $k$, $(X_{n,1},\ldots,X_{n,k})\Longrightarrow(X_1,\ldots,X_k)$ where $X_1,\ldots,X_k$ are i.i.d. with mean zero and variance $\sigma^2$, will the array inherit the CLT ...
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1answer
50 views

Convergence of discretized process when its predictable part converges to infinite variation process

This question seems to be related to Theorem IX.7.28 in J. Jacod and A. Shiryaev's Limit theorems for stochastic processes (2013), and it is very important to prove asymptotic properties of my ...
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1answer
210 views

How to prove the convergence of this kind of sequence?

$$ x_{n}=\sum^{n-1}_{i=0} {a_i x_{n-1-i}} $$ where $$ \sum^{+\infty}_{i=0} {a_i}=1,1>a_i>0,1>x_i>0 $$ In fact, the specific problem (comes from probability theory) I want to solve is that: ...
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120 views

On the remainder of a power series evaluated on the boundary of its convergence disk

Background This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...
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85 views

Sequence of minimal surfaces with bounded second fundamental form and area

Let $M^3$ be a closed orientable smooth manifold, let $g_n$ be a sequence of Riemannian metrics on $M$ converging to $g$ and let $\Sigma_n$ be a sequence of closed orientable $g_n$-minimal surfaces ...
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1answer
107 views

Convergence of the solutions of a ODE system

Consider this system of differential equations for $t\in[0,\infty)$: $$ \frac{d}{dt}x(t) = a(t) + F(x(t), y(t)),$$ $$ \frac{d}{dt}y(t) = a(t) + G(x(t), y(t)),$$ with positive initial conditions: $y(0)&...
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76 views

$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]

For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is $$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
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1answer
42 views

Convergence of localic maps

We can define a limit of a sequence of points in a locale in the usual way: $x$ is a limit of $\{ x_i \}_{i \in \mathbb{N}}$ if, for every open $U$ containing $x$, there exists $N$ such that $x_n$ ...
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108 views

$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?

For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold? $$ \left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right) $$ I ...
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1answer
157 views

Hamming distance between $a+b$ and $a \oplus b \oplus ((a \land b) \ll 1)$

Motivation. In their paper about the cryptographic scheme NORX, the authors use a fast approximation of + by bitwise operations (taking fewer CPU cycles than proper addition) using the formula $$a+b "=...
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132 views

Normal distribution by successive approximation?

$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
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2answers
660 views

What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
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1answer
188 views

Is there an asymptotic bound between converging and diverging series? [closed]

Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$, $$ \log^{[k]}(x) = \begin{cases} \log^{[k-1]}(\log(x)) & k>0 \\ x & k=0 \end{cases}. $$ It is well known, ...
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84 views

Banach fixed point theorem / convergence squeeze

I am trying to prove a convergence result on an iterative scheme which has the initial point defined as $$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$ where s(x) is some unknown function. Here is my theorem and ...
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63 views

Convergence of random operators

I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
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103 views

Inductive limit of inclusions

Let $(\Lambda, \le)$ be a directed system and $\{ X_{\alpha} \}_{\alpha \in \Lambda}$ be a family of topological spaces indexed by $\Lambda$ such that $X_{\alpha} \subseteq X_{\beta}$ whenever $\alpha ...
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1answer
136 views

Limit points and Homeomorphism

I was asking this question at Mathematics SE but I got nothing at all. This is why I am trying this site. We consider the topology of the extended real line. Let $h\colon [-\infty,\infty]\to\Bbb R$ ...
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34 views

The rate of convergence of Markov chain to stationary distribution

Let $X_t$ is Markov chain with transition rates $c: G \times G \rightarrow [0: +\infty)$, where $c(x, y) > 0$, $c(x, x) = -\sum_y c(x, y)$ for $x \neq y$. If $\mu_t(x)$ is the distribution of chain ...
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49 views

Conditional moments estimate

Let $V\in C^\infty(\mathbb R^{d_1}\times \mathbb R^{d_2};\mathbb R)$, such that $Hess V(x,y)\geq \alpha\,I$ for some $\alpha>0$ (namely $V$ is uniformly convex). Thus $V$ as a unique minimum point $...
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1answer
148 views

Functions with at most linear growth at infinity: is the constant itself continuous?

I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M_f$ such that \begin{...
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130 views

Relationship between Hausdorff convergence of sets and indicator functions

Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.: $$ \lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\...
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1answer
282 views

Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$?

$$F(m,n)= \begin{cases} 1, & \text{if $m n=0$ }; \\ \frac{1}{2} F(m ,n-1) + \frac{1}{3} F(m-1,n )+ \frac{1}{4} F(m-1,n-1), & \text{ if $m n>0$. }% \end{cases}$$ Please a proof of: $$\lim_{...
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0answers
233 views

Is the arithmetic-geometric mean of 1 and 2 rational?

It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
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183 views

Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n)} =\lim_{n\rightarrow \infty}\frac{F(n+1,n)}{F(n,n)} $?

Let $\alpha,\beta, \gamma \in \mathbb{R}^+$ be and the function $$ F(m,n)= \begin{cases} 1, & \text{if $m n=0$ }; \\ \alpha F(m ,n-1)+ \beta F(m-1,n )+ \gamma F(m-1,n-1), & \text{ if $m n>...
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1answer
106 views

Convergence of quadratic form $y^T Q y$ where $y$ is a random iid sequence of length $n$ and $Q$ is an $n \times n$ random matrix independent of $y$

For each positive integer, let $Q_n=(q_{i,j})_{i,j \in [n]}$ be a random $n \times n$ psd matrix. In the limit $n \to \infty$, suppose the eigenvalues of this sequence of matrices are uniformly ...
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1answer
48 views

Covariance in the limit of random variables

Suppose $\{X_n\}$ and $\{Y_n\}$ are two sequences of random variables and we know that $X_n \overset{L^2}{\to} X$ and $Y_n \overset{L^2}{\to} Y$, where $\overset{L^2}{\to}$ means converge in mean ...
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3answers
989 views

What is the limit of $a (n + 1) / a (n)$?

Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise. What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
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22 views

Convergence mode with inputs and functions varying in tandem

Given a sequence $(f_n)$ of functions between metric spaces, let's say that $f_n$ "converges flexibly" to $f$ if, whenever $x_n \to x$ is a convergent sequence of inputs, it follows that $...
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130 views

Limiting behavior of a sequence of polynomials

Let $f(z)\in\mathbb{C}[z]$ have all its zeros on the line $\Re(z)=\alpha$ for some $\alpha\in\mathbb{R}$. It is an elementary fact (equivalent to Lemma 9.13 here) that if $u\in\mathbb{C}$ and $|u|=1$, ...
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1answer
118 views

Does convergence of convex sets in Hausdorff distance implies convergence of the complementary sets?

Definition: The Hausdorff distance associated with a distance $d$ on a space $E$ between two sets $A\subset E$ and $B \subset E$ is $d_H(A, B) = \max(\sup_{x\in B}\{d(x, A)\}, \sup_{y\in A}\{d(y, B)\})...
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1answer
68 views

Rate of variance's decrease for the mean's distribution of infinite variance i.i.d. random variables

Consider a set of i.i.d. (positive) random variables $\{X_i\}_{i=1}^N$. Each variable $X_i$ has a distribution with finite mean but infinite variance. In particular, if $P_{X_i}(x)$ is the P.D.F. of ...
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1answer
433 views

Is the harmonic series worse than any summable series?

It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values. We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...
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2answers
100 views

Convergence radius of double series with Pochhammer symbols

I would like to know the convergence radius of the following two double power series of $(x,y) \in \mathbb{C}^2$: \begin{align} \sum_{m,n=0}^\infty \frac{(d-a)_{n+m}(d+b)_{n+m}(d+a)_n(d-b)_n}{n!m!(2d-...
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1answer
52 views

$\|1_{\{f>M\}}f\|_{L^\infty_t L^p}\rightarrow 0$ as $M\rightarrow \infty$ for $f\in L^\infty([0,T],L^p)$?

Let me be more specific. Let $T>0$ be a finite real number. We know that if $f\in C([0,T],L^p(\mathbb{R}^N))$ is complex valued then the statement $\|1_{\{|f|>M\}}|f|\|_{L^\infty([0,T], L^p(\...
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0answers
46 views

Convergence of numerical method [closed]

I would like to prove that the following sequence : $S^{n+1} =1 - I +\frac{β}{α}\ln(S^{n})$, where $\alpha,\,\beta,\,I$ are constants and $S^{0} = 1$ converges as long as $\alpha\cdot S<\beta.$
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3answers
452 views

Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$

Problem: Let $x_1 = 1$ and $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2, \ n\ge 1$. Find the third term in the asymptotic expansion of $x_n$. I have posted it in MSE six months ago without ...
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0answers
53 views

Applying Tannery's theorem to generalised hypergeometric functions

I am thinking about applying Tannery's theorem to some generalised hypergeometric functions, which seems to be a standard method to derive various formulæ. For example, \begin{eqnarray} \lim_{n\to+\...
1
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1answer
261 views

About generalized continued fractions

Let us consider the sequences $(x_n), (a_n)$, starting with $n=0$ and $x_0\in ]0,1[$, defined by the following generalized Gaussian map: $$x_{n+1}=\frac{\lambda_n}{x_n^{\alpha_n}}-\Big\lfloor \frac{\...
0
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1answer
213 views

Sufficient conditions for an asymptotic compactness

This question relates a theory of Mosco convergence. Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$. A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
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1answer
64 views

Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?

The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions : Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...

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