Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

2
votes
0answers
76 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
71 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
92 views

When does almost sure convergence of stochastic integral imply $L^2$ convergence?

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes. ...
2
votes
0answers
28 views

How to get the limit of the equation?

I have got the equation as follow, $\lim\limits_{n\rightarrow \infty}\prod\limits_{j=1}^{n}\left(1+ \dfrac{\mathit{r_j} \cdot {u_j}}{{|{v_j}|^{1/\beta}}}\right)$ where $r\sim N(0, 1)$, $v\sim N(0, ...
2
votes
0answers
37 views

Prove that the sequence $1+\sum_{k=1}^{n}\frac{k+1}{3^{k}+1}$ converges using Cauchy

I need some help with a homework question i'm having difficulty with. Here is the question: "Use the definition of cauchy sequence to prove that the series ...
2
votes
0answers
63 views

Understanding the difference between uniform convergence , convergence everywhere and almost sure convergence

I'm trying to get the intuition behind convergences . I understand convergence in distribution and convergence in mean-square error. However I'm vague about the difference between - almost sure ...
2
votes
0answers
34 views

The series of reciprocals of the integers that do not contain 9 in their decimal representation

Does the following series converge or diverge? $\sum_{n=1}^{\infty} a_n$ where $a_n = \frac 1 b_n$, and $(b_n)_n$ is the subsequence of $(n)_n$ whose terms do not have a $9$ in their decimal ...
2
votes
0answers
52 views

Convergence a.s. implies convergence in $L^{1}$

Suppose $\lim_{n \to \infty}X_{n} = X$ almost surely. Let $Y=\sup _{n}|X_{n}-X|$. Show that $Y < \infty$ almost surely, and define a new probability measure $Q$ by ...
2
votes
0answers
32 views

Radius of Convergence and Interval of Convergence: Am I doing it right?

I must find ROC and IOC in $$ \sum_{n=1}^\infty \frac{{(-1)}^n(x^{2n})}{\sqrt[3]{n^2+4n}} $$ I get $$ R= \lim_{n\rightarrow \infty}\left| \frac{a_n}{a_{n+1}} \right| = \cdots = \lim_{n\rightarrow ...
2
votes
0answers
73 views

The Bessel Function Bessel$J_0(\sqrt{2x})$ is uniformly convergent on $[x_0-1, x_0+1]$

Please note that this is a "follow up" of my previous question $f(x)=\sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n $ is continuous. I decided to accept the answer given there ...
2
votes
0answers
48 views

Convergence of $\sum_n^\infty (-1)^n\frac{\sin^2 n}n$ , questions

Could anyone give a hint how to prove the convergence of the following sum? $$\sum_n^\infty (-1)^n\frac{\sin^2 n}n$$ I tried writing it like this instead: $$\begin{array}{lcl}\sum_{n=1}^N ...
2
votes
0answers
207 views

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{ 2^2}+\cdots+\frac{1}{n^2}$

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{2^2}+\frac{1}{3^2}+ \ldots +\frac{1}{n^2}$ My attempt Take $|x_m-x_n|$, where $m>n$, We have ...
2
votes
0answers
108 views

What's convergence in a group look like?

How should we define convergence for sequences and series in groups? Here's maybe how to do it: Let $G$ be a group. A norm will be like a norm defined on a vector space except we'll define it with ...
2
votes
0answers
20 views

Decomposability of a function into two

This is question on the existence of a decomposability of a function $f(v)$ into multiplicative factors $\lambda(v)$ and $g(v)$. The question is non-trivial since the functions $\lambda(v)$ and $g(v)$ ...
2
votes
0answers
46 views

Likelihood ratio converges in Prob to Pearson chi square under null

Can you guide me on how I can prove that for a multinomial distribution of the form $Y \sim Mult_k(n,p)$, likelihood ratio converges in probability to pearson chi square under null. In other words, ...
2
votes
0answers
43 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
2
votes
0answers
167 views

Approximating hypergeometric distribution with poisson

I'm am currently trying to show that the hypergeometric distribution converges to the Poisson distribution. $$ \lim_{n,r,s \to \infty, \frac{n \cdot r}{r+s} \to \lambda} \frac{\binom{r}{k} ...
2
votes
0answers
59 views

Criteria for $L^1$ convergence looking at Laplace transforms

Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
2
votes
0answers
52 views

measure theory and convergence

1) Let $\Omega=[0,1]$, $F = B([0,1])$, $P$ be Lebesgue measure on $[0,1]$ ($P([a,b])=b-a$). Set $$A_n^i:=\left[{\frac{i-1}{n},\frac{i}{n}}\right]$$ and $$X_n^i(\omega):=\chi_{A_n^i}(\omega)$$ ...
2
votes
0answers
14 views

Polynomials, integrals convergence

Let $P_n(x)= \frac{x^n(bx -a)^n}{n!}, \ \ \ a,b,n \in \mathbb{N}$. Prove that $\int_0 ^{\pi}P_n(x) \sin xdx \rightarrow 0 \ \ \ \ $ and $ \ \ \ \ \int_0 ^r P_n(x)e^xdx \rightarrow 0$ $ \ \ \ \ \ \ (n ...
2
votes
0answers
46 views

Bounding an implicitly defined sequence

I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined implicitly as $$ \lambda_0 = \frac{1}{2},$$ and $$\lambda_{k+1} = \max_{\lambda\in[1,b]} \left\{\frac{1}{2\lambda}\prod_{0\leq ...
2
votes
0answers
75 views

When $\ell^2$-convergence implies $\ell^1$- convergence?

Consider a sequence $(x_n)_{n\in\mathbb N}$ in $\ell^1$ (sequences taking their values in $\mathbb R$), where $x_n=(x_{i,n})_{i\in\mathbb N}$. What are sufficient conditions on the sequence ...
2
votes
0answers
36 views

Convergence of $c_j = \sum_{k=1}^nx_k\cos(2\pi\frac{k-1}{n})j$

S. Kim, K, Umeno, and A. Hasegawa, Corrections of the NIST Statistical Test Suite for Randomness (available at http://arxiv.org/pdf/nlin/0401040.pdf) mention page 8-9 that: $c_j = ...
2
votes
0answers
60 views

convergence in metric space

Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
2
votes
0answers
38 views

Power series dense set of points of convergence

Give an example of power series with dense subset of points on a circle at which the series is convergent and dense subset of points on a circle at which it is divergent. Could you tell me how to ...
2
votes
0answers
41 views

Estimate the error due to replacing the sum of the series with its first n terms

Estimate the error due to replacing the sum of the series with the sum of the first $n$ terms. $1+ \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}-...+\frac{(-1)^n}{n!}+...$ The hint in the textbook says ...
2
votes
0answers
92 views

Monotonic convergence of powers of a stochastic matrix

Let $P$ be a stochastic matrix (nonnegative and each row summing to 1). Assuming that $P^n$ converges to $\textbf{1}\pi$ as $n \rightarrow \infty$, where $\pi$ is a row vector (stationary distribution ...
2
votes
0answers
113 views

Gamma Convergence of functionals on Probability measures

Would be grateful if someone could provide a hint or an appropriate reference for the following. Notation: $\mathcal{P}(\mathbb{R}^n)$- Space of probability measures on $\mathbb{R}^n$ ...
2
votes
0answers
74 views

Limit in norm of a Sobolev space

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\}$ and I have to show that the function ...
2
votes
0answers
58 views

Harmonic series with some numbers deleted for what $\alpha$ is $\sum ' \frac{1}{n^{\alpha}}$.

I have a question about a subseries of harmonic series with reciprocals of natural numbers containing a certain digit deleted. I know how to prove that such series is convergent when we delete all ...
2
votes
0answers
203 views

Central limit theorem - speed of convergence in center vs tails

I've been told that one of the implications of the central limit theorem is that as we increase the sampling of random variables, we converge faster to a normal distribution in the center and slower ...
2
votes
0answers
73 views

Convergence, series with exp and ln

$$\displaystyle\sum_{n=2}^\infty\frac{\exp(n)}{\exp(n\sqrt[n]{n})\ln^{2}n}$$ Tell, if the series converges. Please verify my answer. $$\displaystyle ...
2
votes
0answers
126 views

Scaling function and dilation equation in MRA

In the book I am reading, the author explains how we can use the dilation equation to obtain the scaling function through the process of iteration. In particular, we use the dilation equation, which ...
2
votes
0answers
73 views

Ergodic/stochastic convergence

I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment: The whole assignment deals ...
2
votes
0answers
107 views

uniformly convergent

Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$. Assume that ...
2
votes
0answers
53 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
2
votes
0answers
137 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
2
votes
0answers
59 views

Convergence of integral to convergence a.e question

OK, so now I have $$\int_0^T{\phi(t)h_n(t)} \to \int_0^T{\phi(t)h(t)}$$ which holds for all $\phi \in C^\infty_c(0,T).$ From this how can I deduce that $h_n \to h$ (a.e)? I think I need maybe ...
2
votes
0answers
235 views

Weak convergence and limit of this sequence

Let $f_n$ be bounded uniformly in the $H^1$ norm, so we have (weak convergence) $$f_n \rightharpoonup f \qquad \text{in} \qquad H^1(\Omega\times [0,T]).$$ Then by compact embedding, we have strong ...
2
votes
0answers
49 views

What happens at “spherical shell” of convergence?

In a Taylor series, the convergence/divergence behavior at the boundary case $|x-x_0|=R$ is not immediately determinable. If I understand correctly, such distribution of convergence and divergence ...
2
votes
0answers
116 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 ...
2
votes
0answers
78 views

Alternative way of showing convergence of central moments

I woud like to show for a random variable $X$ $${1\over n} \sum_{i=1}^n (X_i-\bar{X}_n)^q\to E(X-EX)^q \quad (\text{convergence in probability)}$$ My approach was to define $Y_i:= (X_i-\bar{X}_n)$ ...
2
votes
0answers
205 views

Complex Analysis: Power Series and Convergence

Let $f(z)=\sum_{n=0}^{\infty}c_nz^n$ have radius of convergence $R$. Problem Prove that $\sum_{n=0}^{\infty}\overline{c_n}z^n$ has radius of convergence $R$ and that ...
2
votes
0answers
117 views

Uniform Convergence of $a_n$ given its decreasing and converges pointwise

Let $(a_n)$ be a decreasing sequence of continuous functions on the interval $[a,b]$ which converges pointwise to $0$. Show that it converges uniformly to $0$. This is my attempt at the proof, could ...
2
votes
0answers
363 views

Finding a radius of convergence

Let $\sum_0^{\infty} a_n z^n$ have radius of convergence $R$ with $0< R< \infty$. Let $\alpha>0$. Find the radius of convergence of $\sum_0^{\infty} |a_n|^{\alpha} z^n$. I tried to start ...
1
vote
0answers
21 views

Series convergent but not absolutely? $\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$

For which real numbers $p>0$ does the series $$\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$$ converge? Obviously it converges absolutely for $p>1$ but what about $0<p<1$? I have the ...
1
vote
0answers
36 views

Gauss' test for Convergence

In the text I am using, the hypotheses used for Gauss' test for convergence are different to others I have seen. The text has the hypotheses: if the series $\sum_{n=1}^{\infty} a_n$ is such that ...
1
vote
0answers
27 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 ...
1
vote
0answers
25 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow ...
1
vote
0answers
42 views

Change of signs in harmonic series

Prove that if we choose signs for individual terms in harmonic series $\sum_{n=1}^{\infty}{1\over n}$ in such a way that $p$ positive terms are followed by $q$ negative terms (without rearranging the ...