Convergence of sequences and different modes of convergence.

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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 ...
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22 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 ...
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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 ...
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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.
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12 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 ...
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1answer
31 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 ...
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1answer
24 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 ...
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1answer
15 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 ...
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2answers
31 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 $n→∞$ ...
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39 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 ...
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15 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 ...
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1answer
16 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 ...
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1answer
11 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
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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 ...
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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 ...
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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 ...
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36 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} ...
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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 ...
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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 ...
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14 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 ...
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1answer
26 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: ...
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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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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
27 views

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

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 $) ...
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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 ...
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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 ...
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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? ...
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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 ...
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4answers
51 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 = ...
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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 ...
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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 ...
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2answers
52 views

Find the radius of convergence of a power series.

Consider the power series $$\sum_{n=0}^{\infty}a_nx^n$$ where $a_0=0$ and $a_n=\frac{\sin(n!)}{n!}$ for $n≥1$ Let $R$ be the radius of convergence then $R≥1$ $R≥2π$ $R≥π$ My attempt: I used the ...
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1answer
22 views

Convergent Sequence and its limit

Can anybody help me out in this problem: I am not able to figure out how the value of lambda in the 2nd problem comes out to be 2 ? In the first problem value of lambda came out to be 2 after ...
1
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1answer
29 views

Explicit example of normal family

Suppose $\mathscr F \subset H(\Omega$) for some region (i.e. open connected) $\Omega$. ($H(\Omega)$ means the set of all holomorphic function in $\Omega$) We call $\mathscr F$a normal family if every ...
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3answers
65 views

Determine wether the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ diverges or converges? - I want to check if my reasoning is correct

I got that the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ converges by doing the following: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ = $\sum_{n=0}^\infty ...