Convergence of sequences and different modes of convergence.

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3
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140 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 ...
3
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157 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 ...
3
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157 views

Hölder Condition Implying Uniform Convergence

Define $g(z)=\frac{1}{2\pi i}\int_{-k}^k \frac{h(\zeta)}{\zeta-z}d\zeta$, where $h$ is continuous and defined on $[-k,k]$. Let $|h(x)-h(y)|\leq |x-y|^\alpha$ for all $x,y\in[-k,k]$ and for some ...
2
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78 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
2
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35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
2
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50 views

Find the limits of the convergent subsequences

Let my sequence be $a_n=n\pi-\lfloor n\pi\rfloor$ This sequence is bounded in $[0,1)$ so if must have a convergent subsequence. In fact, it seems to me like it has infinitely many convergent ...
2
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23 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
2
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97 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
2
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32 views

problem on almost sure convergence

Let {$X_i$} be iid with finite second moment. Let $Y_n = \frac {2} {n(n+1)} \sum_{i=0}^n i*X_i $, n$\ge$1 Show that $Y_n \to E(X_1) $ I tried to define $Z_i = \frac {2} {(n+1)} i*X_i $ Then $Y_n = ...
2
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31 views

Question on the Raabe's test

Bartle - Real analysis Raabe's Test Let $\{x_n\}$ be a sequence in $\mathbb{R}$ If there exists $a>1$ and $K\in\mathbb{N}$ such that for all $n\geq K$, ...
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22 views

A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
2
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50 views

Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
2
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42 views

fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle ...
2
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33 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
2
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44 views

When to Interchange Limit & Integral

I got stuck in the proof of Cauchy's Integral Formula for higher derivatives: Under what Conditions over a function $f$, we can infer that : $\displaystyle\lim_{h\rightarrow ...
2
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37 views

Does this condition imply convergence of $\sum_k x_k$?

Let $\{x_n\}_{n \geq 1}$ be a sequence of real numbers (in principle, they could be complex numbers, but I don't think this makes much of a difference for this problem). Suppose the sequence $x_n$ ...
2
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50 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 ...
2
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19 views

Convergence of Iteration with Sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and the iteration $$ x_{k+1} = \frac{1}{k} \sum_{i=1}^{k} f( x_i ) $$ for some given initial condition $x_1 \in ...
2
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42 views

False convergence result

I wrote a proof below that if A(n) is decreasing and lim A(n) = 0 for some sequence, then the corresponding series must converge. I know this is false, with the harmonic series as a counterexample, so ...
2
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70 views

Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...
2
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89 views

Uniform Convergence of $f_n(x)=1+\cos(x/n)$

How to see that the series of functions $f_n(x)=1+\cos(x/n)$ does not converge uniformly on $\mathbb{R}$. To prove $f_n(x)$ does not converge uniformly to $1$ is easy. Let $x=0$, then $|f_n(0)-1| = 1 ...
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150 views

Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
2
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36 views

Are all functions in a Banach space convergent?

Are all functions in a Banach space convergent? I need this answer in a study of wavelet analysis. My thoughts are: since we have this definition: Let $X$ be a Banach space. A sequence of vectors ...
2
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118 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
2
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26 views

Compactness in topology of uniform conergence (of functions and all their derivatives) on compact subsets of (0,\infty)

I am trying to understand an example in the book "Lectures on Choquet's Theorem" (R.R. Phelps). My question is: Given the space of real valued infinitely differentiable functions on $(0, \infty)$ ...
2
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58 views

Help with Series Convergence

Can someone help me prove that the following series $S$ converges: $$S=\sum_{m=1}^\infty\frac{1}{m^2|\sin(m)|}$$ I would appreciate any help in constructing a simple proof.
2
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80 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)$ $ ...
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76 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
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126 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. ...
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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
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42 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
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85 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
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37 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 ...
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58 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
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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 ...
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81 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 ...
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49 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
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250 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
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109 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 ...
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21 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 $g(v)$ ...
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193 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
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72 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
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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
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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 ...
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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
79 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
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37 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
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66 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
40 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
45 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 ...