Convergence of sequences and different modes of convergence.

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3
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53 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
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76 views

$\sum\limits_{n=1}^{\infty}\frac{\sin(nx)}{n^{\alpha}}$ does not converge uniformly for $\alpha \in (0,1]$

The question is, why $\sum\limits_{n=1}^{\infty}\frac{\sin(nx)}{n^{\alpha}}$ does not converge uniformly for $\alpha \in (0,1]$. I have a solution, but I dont understand the estimates, sorry=(. ...
3
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42 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
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26 views

Product over a factorial convergences

I'm working on a problem and need this to converge to any value: $\displaystyle \frac{(1/2)((1/2)-1)((1/2)-2)\cdots ((1/2)-n+1)}{(n+1)!} = \Pi_{j=1}^{n} \frac{\frac{1}{2} - j+1}{j+1}$ The ...
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82 views

Weak vs strong convergence for unitary operators

Suppose $H$ is a separable complex Hilbert space with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$, where $\|u\|^2 = (u,u)$. Suppose $u, u_1, u_2, \dots \in H$. Then $\lim_{n \to \infty} u_n = ...
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84 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
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46 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
3
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53 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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30 views

Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...
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51 views

Does $L^{1}$ convergence implies almost everywhere cesaro convergence?

Let $X$ be a compact metric space, $\mu$ a Borel measure and $f_{n} \in L^{1}(X,\mu)$ continuous functions such that $f_{n}(x) \overset {L^{1}} \rightarrow 0 $. Now can we deduce that ...
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36 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
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45 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
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74 views

snails on a tetrahedron

suppose you have $4$ snails arranged on the surface of an equilateral tetrahedron(all sides equal, each snail is numbered from $1$ to $4$, and that the snail numbered $1$ moves toward snail numbered ...
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58 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
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41 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
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49 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
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91 views

Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
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170 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
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72 views

convergence of series implies convergence of coefficients

Is it true that $$\sum_{i=0}^\infty a_{i_n} y^i \rightarrow \sum_{i=0}^\infty a_{i} y^i \quad \forall y \in [0,1]$$ implies $$a_{i_n} \to_{n \to \infty} a_{i} \quad \forall i$$ where $0 \leq ...
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354 views

derivative of limit function vs limit of derivatives

Suppose that we have a sequence of differentiable functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ such that $f_n$ converges to some function $f$. Then it is not necessary that the sequence of ...
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67 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
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110 views

maximum renewal rate of a Markov chain

Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ ...
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124 views

sequence convergence

Assume all terms $a(n) >0$ , $\sqrt{a(1)}\geqslant 1 +\sqrt{a(0)}$, and $$\left|\dfrac{a(n+1)}{a(n)}-\dfrac{a(n)}{a(n-1)}\right|\leqslant \dfrac{1}{a(n)} $$ for all $n>0$. Prove that ...
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201 views

Cluster point of $a_{n}:=n+(-1)^{n}n$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution? The subsequence diverges for increasing even $n$ since $2n$ grows infinitely. The ...
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144 views

Infinite product convergence

Reading some notes I have taken during class, I am unable to see the connection between these lines: $g(z)=\prod_{k\geqslant 0}(2z\lambda_k + 1)$, where $\lambda_k=4/((2k+1)^2 \pi^2)$ And The zeroes ...
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190 views

Cauchy criterion

We know that the series $\sum_{n=1}^\infty \dfrac{1}{n(n+1)}$ coverge to 1. I want use Cauchy criterion to show the series converges as an exercise. Is the following proof correct? Given $\epsilon ...
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61 views

The existence of the limit $\lim_{n\to\infty} S_{i_n}\circ…\circ S_{i_1}(x)=x_0$ for some $i_1, i_2,…\leq N$

Let $(X,\rho)$ be a Polish space. We are given continuous functions $S_i: X\rightarrow X$ ($i=1,...,N$) and a constant $0<r<1$. We assume the following: For each $x,y \in X$ there exists ...
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275 views

Convergence of sequence in $C^{k, \alpha}$ composed with $C^\infty$ function

Is it true that if a sequence $u_n \to u$ in $C^{k, \alpha}$ norm, and if you have a function $f \in C^\infty$, then $f(u_n) \to f(u)$ in $C^{k, \alpha}$? I think so, since this is true for ordinary ...
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424 views

Uniform convergence of analytic functions

Let $f_{n}(z), g(z)$ be entire functions, for all $n\geq 1$. Suppose that $g(x)$ doesn't vanish on $\mathbb H\cup\mathbb R$ (so we have $\frac{f_{n}(z)}{g(z)}$ analytic on $\mathbb H\cup\mathbb R$). ...
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138 views

Another Question based on Abel's theorem of multiplication of series

I am trying to show that the $qth$ power of the series $$a_{1}\sin \theta +a_{2}\sin 2\theta +\ldots +a_{n}\sin n\theta +...$$ is convergent whenever $q(1-r)<1$, r being the greatest number ...
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194 views

How to prove that $\lim_{\lambda \rightarrow 0} f*h_\lambda (x)=f(x) $ a.e.

Assume that $h_\lambda(x)=\frac{1}{\pi} \frac{\lambda}{\lambda^2+x^2}$, for $\lambda>0$, $x \in \mathbb{R}$. I know that if $f\in L^p$ then $\lim_{\lambda \rightarrow 0} \|f*h_\lambda -f\|_p =0$, ...
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171 views

Uniform convergence of measures

Let $A\subset[0,1]^n\subset\mathbb{R}^n$ be a closed semi-algebraic set. Let $f_k: [0,1]\rightarrow\mathbb{R}$, $f_k(x_i)=\mu^{n-1}(A_{|_{x_i}}+B^{n-1}_{1/k})$ where $A_{|_{x_i}}=A\cap H^i_{x_i}$ ...
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166 views

Convergence of marginal distribution in the case of convergence in distribution

Suppose a random variable $Z_n=(X_n,Y_n)$ takes a value on $A= [0,1] \times B=[0,1]$. We can consider a conditional distribution of $Y_n$ given the realized value of $X_n$. Now suppose $Z_n$ converges ...
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176 views

Uniform convergence of a series

This problem came from the Krantz text ($2^{nd}$ ed. ch. 9, prob. 17): Prove that the series $\displaystyle\sum_{j=1}^{\infty }{\frac{\sin{(jx)}}{j}}$ converges uniformly on compact intervals that do ...
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38 views

Limit of a function is the function of the limit (no continuity or convergence)?

Suppose I have a sequence of real-numbers $\{a_n\}_n$ and that $\lim_{n \rightarrow \infty}a_n=a$ where $a$ can be finite or infinite. Consider a function $f(\cdot)$ of $a_n$. Under which conditions ...
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48 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
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21 views

Is $\sum \frac{n+1}{b_{n+1}}$ irrational, when $b_1=2$ and $b_{k+1}=2^kb_k(b_k-1)+1$, $k\geq 1$?

Let the sequences of positive integers $$a_n=n$$ when $n\geq 1$, and $$b_{n+1}=2^nb_n(b_n-1)+1$$ for $n\geq 1$ taking $b_1=2$. I've computed with previous sequences to assert that satisfy the ...
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35 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
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37 views

Measure Theory - understanding usage of Dominated Convergence Theorem?

I am trying to understand the given proof below. But I don't really understand how the Theorem of Dominated convergence is applied? Which is the function that "dominates" the sequence? and we need ...
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26 views

Convergence of infinite-dimensional random variables

For every $n \in \mathbb{N}$ and every measurable $E \subseteq [0,1]$, the object $f_n(E)$ is a random variable that takes real values. The sequence ($f_n$) can thus be understood as a sequence of ...
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51 views

Prove $|x_{k+1}-x_{*}|\le \rho|x_k-x_{*}||x_{k-1}-x_{*}|$ is equivalent with $|x_{k+1}-x_{*}|\le \delta|x_k-x_{*}|^p$, where $1<p<2$.

The definition of convergence in super-linear is $$|x_{k+1}-x_{*}|\le \rho|x_k-x_{*}||x_{k-1}-x_{*}|$$ I am asked to prove this is equivalent to $$|x_{k+1}-x_{*}|\le \delta|x_k-x_{*}|^p$$ where ...
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17 views

Stochastic Convergence

I need help figuring out if a series of "apparently random" digits are the result of the same (possibly non-polynomial) function, ergo, not-random but deterministic. The highest level math I know is ...
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40 views

Is this sequence of function dominated?

Let $$f_n(t) = \left\{ \begin{array} {lr} t^k e^{-t} \left( 1 - \frac {e^{-t}} n \right)^{n-1}, & t > -\log n \\ 0, & t \le -\log n \end{array} \right.$$ for $k=1,2$. Is there an integrable ...
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70 views

The series of $\sin A$, when $A$ is a $n \times n$ matrix

I have to solve this problem for university. $\sin A$ is defined for every $n \times n$ matrix $A$ as: $$\sum_{n=0}^\infty \frac{(-1)^n A^{2n+1}}{(2n+1)!} \tag{$\star$} $$ Prove that a) $\| \sin ...
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47 views

Does this series converge? If so, to what?

i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am ...
2
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47 views

show that the following integral converges

I want to show that $f(a) = \int _ 0 ^\infty \frac {\sin ( x^2 + ax )}{ x } dx$ converges for all $a \ge 0$ and $f$ is continuous on $[0, \infty ) $. I don't really know how to start. I know $\sin ...
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25 views

Almost sure convergence of eigenvectors under nice assumptions

I have that a sequence of symmetric, real, positive semi-definite random matrices, $M_n$, converges almost surely to a real-valued positive semi-definite diagonal matrix, $D$, with at least one ...
2
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25 views

Interpretation of different convergence results for random series

Let $X_k, k\geq 1$ be a sequence of random variables and let $S_n:=\sum_{k=1}^n X_k, n \geq 1$ be the sequence of partial sums. When the $X_k$ are Independent, Kolmogorov's 3-series Theorem gives ...
2
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37 views

Test integral convergence

I'm given the integral $$\int \limits_0 ^{+ \infty} \frac{e ^ {-\cos t} \cdot \sin (t ^ \beta)}{t^\alpha} dt \qquad a,b \in \mathbb{R}$$ and I need to test the absolute convergence. I split it in two ...
2
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28 views

Is the set of linear combinations dense in the set of the dual space of $l_p$?

Good day, Right now I'm working with the book "Functional Analysis" by Bachman and Narici, it is available on Google Books, see ...