Convergence of sequences and different modes of convergence.

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2
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1answer
13 views

Cauchy sequence of partial sums of orthogonal vectors in a general Hilbert Space.

Let $(x_n)$ be a sequence of orthogonal vectors in a Hilbert space (V, <,>). For $n = 1, 2, 3, ... $ put, $$s_n = \sum_{j=1}^{n} x_j.$$ (a) Calculate $||s_n||$ in terms of ...
0
votes
0answers
16 views

Convergence in normed linear spaces $\ell^p$ and $\ell^1$

I have gone through a proof that shows that if $(x_n)$ is a sequence in $\ell^p$ for $1<p<\infty$, and $x\in \ell^p$ such that $x_n(k)$ converges to $x(k)$ for each $k$ and $(x_n)$ is uniformly ...
-2
votes
0answers
13 views

Convercenge in probability implies convergence in Lp [on hold]

Show that if $X_n$ is that $|X_n|< C$, with $C\in \mathbb{R}$, $\forall n \in \mathbb{N}$, then $X_n \overset{P}{\rightarrow} 0 \implies X_n \overset{{L^P} }{\rightarrow}0$
0
votes
1answer
11 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
1
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0answers
10 views

Understanding the difference between convergence in distribution and convergence almost surely

I know that the sum of $\sum_{i=0}^nZ_i$ where $Z\sim N(0,1)$ has a distribution of a Chi squared distribution with $n$ degrees of freedom which in my understand means that $Z^2$ converges in ...
2
votes
1answer
33 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
2
votes
1answer
16 views

Pointwise convergence - $\frac{nx}{1+n \sin(x)}$ , $x \in [0, \frac{\pi}{2}]$

Is anyone able to check if this is correct: for $$f_n(x) = \frac{nx}{1+n \sin(x)} , x \in [0, \frac{\pi}{2}]$$ Does this converge pointwise to $$ \frac{x}{\sin(x)}$$ I am unsure due to the fact ...
-3
votes
1answer
22 views

Show, directly from the definition, that the following series is convergent. [on hold]

Using the definition of a convergent series, how do you show that the series $\sum_{n=1}^{\infty} (\frac{-2}3)^n $ converges.
1
vote
0answers
15 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise if $f:[0,1]\to \mathbb R$ is only piecewise continuous (but ...
3
votes
1answer
33 views

Is it true that $\sum_{n=1}^\infty \dfrac{Var(Y_n)}{n}<\infty$?

Suppose that $\{X_n\}$ is an i.i.d. sequence of random variables with $E|X_1|<\infty$.Define $Y_n=X_nI_{\{|X_n|<n\}}$ for all $n\geq1$. Is it true that $\sum_{n=1}^\infty ...
1
vote
1answer
27 views

Convergence of $u * \eta_\epsilon$

Let $\eta \in C_c^\infty(B(0,1)), \eta \ge 0, \eta$ radially symmetric and $\int_{\mathbb{R}^n} \eta d\mathcal{L}^n = 1$. $\eta_r := r^{-n} \eta(\frac{x}{r}) \in C_c^\infty(B(0,r))$. Integral of ...
1
vote
1answer
16 views

Show that $X_n/n$ does not converge almost surely

I am generally able to prove that a sequence of random variables $X_n$ converge almost surely to a random variable $X$ by using the following strategy: Take any typical sample point ...
0
votes
2answers
32 views

Can one prove divergence by showing a series has at most one solution for an=0?

Say I have any series, I would think it was enough to show that this series equals 0 at most once to prove it diverges. My logic is, For a series: $\sum a_n →∞$, and diverges, if $a_n≠0$ for $n→∞$ ...
3
votes
0answers
50 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
1
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0answers
16 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
2
votes
1answer
21 views

Convergence of random variables under different probability measures

I have a succession of random variables $X_n$ on $\Omega=[0,1]$ with $X_n=(1-\omega)^n$. I have to prove the convergence almost sure and/or in law in these case: $\mathbb P=\delta_{0}$ $\mathbb ...
1
vote
1answer
12 views

If the second moments are uniformly bounded, does $Y_n$ converge in $L^2$?

Let $\{X_n\}$ be a pairwise uncorrelated sequence of random variables such that there exists a fixed constant $c>0$ such that $E(X_n^2)\leq c$ for all $n\geq1$. Does it imply that for any ...
3
votes
1answer
33 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
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0answers
23 views

If $X_n$ is Geo$(\lambda/n)$, does $X_n$ converge in distribution?

If $X_n$ is Geo$(\lambda/n)$, does $X_n$ converge in distribution? Clearly $\lambda>0$ and for all $n$, $\dfrac{\lambda}{n}\leq1$. Now, $$P(X_n\leq ...
0
votes
1answer
22 views

On the proof of the continuity of the inner product.

I am having problems with the following proof and I need to fill in some details: I understand that continuity is being proven by the sequence definition but I do not get why (a) follows ...
0
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0answers
37 views

An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k \cos(kt) + b_k \sin(kt), \\ a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} x(t) \cos(kt) dt, \\ b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} ...
4
votes
1answer
21 views

How to make a topology out of $N$ that involves convergent / divergent sets.

Let $N$ be the naturals $1, 2, \dots$ Call a subset $A$ of $N$ convergent if the reciprocal sum $\sum_{a \in A} \frac{1}{a}$ converges. Similarly call as set divergent if the sum diverges. Notice ...
0
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0answers
23 views

If a bounded sequence is equicontinuous, it has a uniformly convergent subsequence

I am currently having some difficulty with problem 2.7.8 in Introduction to Topology by Theodore Gamelin and Robert Greene. The problem goes as follows A family F of real-valued functions on a ...
2
votes
0answers
15 views

Good Problem Book for Convergence Concepts in Probability

I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are: Borel-Cantelli Lemmas Modes of Convergence and ...
2
votes
1answer
27 views

Normal convergence: $\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$

I want to prove that: $$\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$$ is not normally convergent on $[a, \infty)$ for fixed $a>0$. Let $U_n(x)$ denote the general term. We have: ...
0
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0answers
16 views

Skorokhod vs Meyer zheng topology

I am new to the Skorokhod space and I want to know why Meyer-Zheng topology on the space of càdàg functions is weaker than the standard Skorokhod topology. Thanks in advance!
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0answers
23 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
4
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0answers
34 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
0
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0answers
7 views

convergence multivariate normal

If $X$ and $Y$ have asymptotic normal distribution then using Slutsky's theorem $aX+bY$ is also asymptotic normal, can I conclude that the vector $(X,Y)$ is asymptotic bivariate normal? If not, how ...
1
vote
1answer
27 views

Can somebody explain the notation $f \in C^4$

To give some context the full question is: Suppose $f \in C^4$ in a interval containing the root $\alpha$ and that Newton’s method gives a sequence of iterates $\{x_k\}$, $k = 0, 1, 2, \dots$ which ...
1
vote
3answers
43 views

Can I use the Dirichlet's test to prove the convergence of $\sum_{n=1}^N \frac{e^{in}}{n}$?

I am trying to state that $$\sum_{n=1}^\infty \frac{e^{in}}{n}$$ converges. Is it correct that $|\sum_{n=1}^N e^{in}|\leq M$ for every positive integer $N$? I.e use $e^{in}$ as the $b_n$ term in ...
1
vote
2answers
28 views

Series definied by recurrence relation

Define a sequence recursively by $$\large{a_1 = \sqrt{2},a_{n+1} = \sqrt{2 + a_n}}$$ (a) By induction or otherwise, show that the sequence is increasing and bounded above by 3. Apply the monotonic ...
3
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0answers
41 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 ...
0
votes
1answer
25 views

Is it possible to exhibit a collection of sets

Let a subset $D$ of the natural numbers be called convergent or divergent when the associated series $\sum_{d \in D} \frac{1}{d}$ converges or diverges. Define a topology on $\Bbb{N}$ by defining the ...
0
votes
1answer
27 views

Why $N= max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [closed]

Using the proof development strategy used regarding the proposition (for all $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n - L| < \epsilon$ for all $n > N $) ...
0
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0answers
23 views

Sums convergent but not uniformly convergent on [0,1]

Show that both $\sum_{n=1}^{\infty} ({1-x}){x^n}$ and $\sum_{n=1}^{\infty} (-1)^n({1-x}){x^n}$ are convergent on [0,1] but only one converges uniformly. Which one? Why? I was playing around with the ...
0
votes
2answers
60 views

prove or disprove convergence [duplicate]

Im trying to prove or disprove the following , but I am having a hard time. It seems that the statement is true , but I have no idea how to prove it. If \begin{equation*} \sum_{n} a_{n}^2 ...
0
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0answers
32 views

series and dyadics [duplicate]

I'm in Calc II and am unfamiliar with what dyadics is but my teacher said that it's possible to find the sum of $\frac{\cos n}{n^2}$ by using dyadics. Would you mind laying out a step by step? ...
0
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1answer
19 views

convergence of $\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx$

I am trying to study the convergence of $$\int_{x=0}^{\infty}x^{-\frac{M-1}{2}-N}(1-e^{-x})^{M-1}e^{-x}dx,$$ where $M$ and $N$ are positive integers. I've tried some $M$ and $N$, and it seems that ...
1
vote
4answers
53 views

Determine the convergence of sequence $\tan^2(1/n)$

How to determine the convergence of sequence $$ \sum^\infty_{n=1} \tan^2(1/n) $$ I have used the divergence test $$ \lim_{n\to\infty} \tan^2(1/n) = \tan^2(0) = 0 $$ So we can't say if it diverge ...
2
votes
0answers
32 views

Convergence of a Markov Chain to the normal distribution

If $i$ is a state of an irreducible, postive recurrent Markov chain $X$, and $V_n$ is the number of visits to $i$ between times $1$ and $n$, and further $\mu=\mathbb{E}_i(T_i)$ and ...
0
votes
1answer
25 views

Alternating series convergence test

I have a series $\sum_{n=1}^\infty c_n x^n$ where $c \le c_n \le C$. I can determine radius of convergence easily by the root test, but how does one determine convergence for $x = -1$? It is not a ...
0
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2answers
37 views

Finding the radius of convergence of a power series, $\sum_{n=1}^{\infty} a_n x^n$.

I have to detemernine the radius of convergence of the power series $\sum_{n=1}^{\infty} a_n x^n$, where $(a_n)_{n=0,1,2,...}$ is given by $a_n=2-\dfrac{1}{2}a_{n-1}$ with $a_0=2/3$. So far I've ...
0
votes
1answer
28 views

How to find out if a sequence with exponentiation in fraction is convergent

I need to find the convergence of this function: $\sum^{\infty}_{x=1}{\frac{(x+1)^{x^2}}{x^{x^2}2^x}}$ Now my problem is, I have no clue how to do this (I tried the root-test and it did not work ...
2
votes
1answer
39 views

Prove that $\lim_{n\to \infty}\langle \operatorname{erfc}(-nx), \phi\rangle =\langle H_0, \phi\rangle $

Define the error function $\operatorname{erf}(x)$ as: \begin{equation} \operatorname{erf}(x):=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-y^2}dy \end{equation} and ...
0
votes
0answers
15 views

uniform converging functions

If I know that $\sum_M^\infty v(x)$ converges towards $f$, then $$\left| \left( f + \sum_{1}^{M-1} v(x) \right) - \sum_{1}^N v(x)\, \right| = \left|\;f - \sum_{M}^N v(x)\,\right| < \epsilon$$ for ...
0
votes
2answers
52 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
2
votes
4answers
76 views

Convergence of $\sum_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$

I have encountered the following problem: Determine whether $$\sum \limits_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$$ converges or diverges. What I have tried so far: Assume that $a_n = ...
0
votes
0answers
19 views

Is there something wrong with this proof? Convergence of m-tail of series

I wanted to formally proof that the uniform convergence of the $m$-tail of a series of functions implies uniform convergence of the entire series. It made intuitive sense but I did not know any ...
0
votes
1answer
31 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...