Convergence of sequences and different modes of convergence.

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3
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61 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 ...
3
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104 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 $\mathfrak{S}$ ...
3
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123 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 ...
3
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183 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 ...
3
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136 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 ...
3
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173 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 ...
3
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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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240 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 ...
3
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386 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 $\mathbb H\cup\mathbb R$). ...
3
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130 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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190 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 $\lim_{\lambda \rightarrow 0} \|f*h_\lambda -f\|_p =0$, ...
3
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150 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 $A_{|_{x_i}}=A\cap H^i_{x_i}$ ...
3
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148 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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166 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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15 views

Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
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25 views

Formal Expansion of another Expansion

Given a function $f(x)=\sum_{n=1}^{\infty}\frac{c_n}{x^n n!}$, where $c_n$ are constants, we want to find the formal series expansion of the function $g(x)=\exp(f(x))$ in terms of $x$. I want to ...
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27 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...
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13 views

Multivariate Delta Method

If I have a $\sqrt{N}$ asymptotic normal estimator (call it $\boldsymbol{\theta}$, possibly a vector). Say I want to find the asymptotic distribution of $g(\boldsymbol{\theta})$ and suppose ...
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92 views

Show the following series converges to $\frac{1}{e}$.

Let $\tau \in \mathbb{N} $ whereas $\tau$ fulfils the following condition: $$ \sum_{i= \tau}^{n-1} \frac{1}{i} \lt 1\le \sum_{i=\tau-1}^{n-1} \frac{1}{i}$$ Prove that $$\lim_{n \to \infty} ...
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36 views

A characterization of quadratic variation for $L^2$ martingales

I am trying to prove the following statement but I am totally at a loss. Let $(A_t)$, $t \in \mathbb{R}^+$ be an adapted (with respect to the filtration $(\mathcal{F}_t)$) continuous integrable ...
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56 views

On convergence of the series $\sum_{k=3}^{\infty}\binom{k-1}{2}p^3(1-p)^{k-3}$

Given a number $0 \le p \le 1$ calculate if: $\sum\limits_{k=3}^\infty \binom{k-1}{2} p^{3}(1-p)^{k-3} = 1 $ So: $\sum\limits_{k=3}^\infty \binom{k-1}{2} p^{3} (1-p)^{k-3} = \sum\limits_{k=3}^\infty ...
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33 views

Sequences and series of $\tan^n x$

Please help me with this question: Investigate the convergence of the sequence $$\tan x, \quad \tan^2 x, \quad \tan^3 x, \quad \dots, \quad \tan^n x$$ for $x \in (-90^\circ, 90^\circ)$. ...
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30 views

Convergence of derivative in L1

Let $F(x)$ be a distribution function over $\mathbb{R}$ with positive derivative at origin $f(0)$. Let $Q$ be a measure on $\mathcal{B}(\mathbb{R})$. Can we have following results under some ...
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26 views

Wondering about limit comparison test and its application

I have a question about the limit comparison test. In particular if it is valid to use in the question I will post. I am interested in determining wether the series $$\sum_{n=2}^\infty ...
2
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23 views

Proving that a subspace of $L^2$ is closed.

Suppose $Z$ is a random variable on a probability space $(\Omega, F, P)$. $M(Z)$ is the subspace of $L^2$ consisting of all random variables in $L^2$ which can be written in the form $\phi(Z)$ for ...
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18 views

Convergence of time-slice measures to convergence in law

Question: Are there some conditions that allow one to go from finite dimensional distribution convergence to convergence in path space? Motivation: Consider a sequence of measures for e.g. solutions ...
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32 views

Does the sum $\sum_{n=1}^{\infty}{a_nb_n}$ converge(fourier series coefficients)?

Let $f\in H(0,2\pi)$, with inner product $<f,g>=\int_0^{2\pi}{f(t)g(t)dt}$ $S_f=a_0 + \sum_{n=1}^{\infty}{a_ncos(nx)}+\sum_{n=1}^{\infty}{b_nsin(nx)}$, is the fourier series for f. Where ...
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59 views

Continuity of $t\mapsto \mathbb E X_t$

Is the following statement and its proof correct? Do you know this or related results and where I could read more about such things? Lemma: If $X$ is a right-continuous process, and the collection ...
2
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78 views

Show that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$.

Suppose that $\Sigma_{k=1}^\infty a_k$ converges. Prove that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$. Attemtp: ...
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35 views

If $X\geq0$ is a random variable, show that $\lim\limits_{n\to\infty}\frac1nE\left(\frac{1}{X}I\left\{X>\frac{1}{n}\right\}\right)=0$

If $X\geq0$ is a random variable then show that:$$\lim_{n\to\infty} \frac{1}{n} \cdot E\bigg(\dfrac{1}{X}I\bigg\{X>\dfrac{1}{n}\bigg\}\bigg)=0$$ A hint would be most appreciated. I have ...
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32 views

check for convergence of finite series

Let S$_k$ = $\sum\limits_{n=2}^k $ (-1)$^a$ $\frac{1}{\ln n}$ where a:=floor($n^{0.5} $) Is S$_k$ convergent? Is S$_{2k^2}$ convergent? I have just some vague idea how to show that both of them are ...
2
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31 views

Stochastic order of Max

Consider an array $\{\{X_{ni}\}_{i=1}^n\}_{n=1}^\infty$ s.t. for each $i$, $X_{ni}=o_p(n^\alpha)$. What is the order of $M_n=\max_{1\le i\le n}X_{ni}$? I got the following ...
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30 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the ...
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20 views

Almost sure convergence and boundedness

Let $(\Omega, \mathscr{F}, \mathbb P)$ be a probability space and $(X_n)_{n \in \mathbb N}$ a sequence of random variables on that space, which converge almost surely to a constant $c\in \mathbb R$, ...
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47 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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87 views

$f(x)\sim 1/x \implies (1+f(x))^x\to e$, but what family of functions maximizes the speed of convergence from below?

This problem is subordinate to finding out if $$\left(1+\frac{\log p_{n+1}}{p_n}\right)^{p_{n+1}/\log p_n},$$where $p_n$ is the $n$-th prime, never stabilizes above or below its limiting value, which ...
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24 views

Convergence of sum of exp. decaying pdf // When does L^2 convergence imply a.s. convergence?

The problem: Let $X_t^i$ $(i \in Z)$ be integer valued random variables on the same probability space. Let $m: Z \rightarrow R $ be a symmetric probability density on the integers which has ...
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60 views

the continuity of argmin

how can I show the minimum of a converging sequence of continuous functions is converging to the minimum of the limit of that sequence.i.e. $$\lim_i \text{argmin}_{x} f_{i}(x)=\text{argmin}_{x} \lim_i ...
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45 views

Linear functional norm

We are given $T:l_2\rightarrow \mathbb{R}$ such that $T({x_n})=\sum_{n=1}^\infty \frac{1}{\sqrt{n}}x_{3n}-\sum_{n=1}^\infty\frac{1}{n}x_{2n}$. I would like to know what the norm of that functional is ...
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66 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
2
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30 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 ...
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26 views

Convergence of a kinda bump function to Dirac delta

I'm a bit stuck with the following question. I'd like to know if it is true that $$f_\epsilon(\theta):=\frac{M_\epsilon(\theta)}{\int_{-\pi/2}^{\pi/2}M_\epsilon(\alpha)d\alpha}$$ where ...
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64 views

Convergence of Expectations (cont'd)

The question is related to this question. Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ ...
2
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100 views

Semigroups: Entire Elements (I)

Problem Given a Banach space $E$. Consider a C0-semigroup: $$T:\mathbb{R}_+\to\mathcal{B}(E)$$ Define its generator by: $$Ax:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)\in E$$ (It is a densely-defined ...
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53 views

When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and ...
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41 views

Convergence in the space $C([a,b],M_1(\mathbb R))$.

Let $M_1(\mathbb R)$ be the space of probability measures on $\mathbb R$ with the weak(-*-)topology: $\mu_n \rightarrow \mu$ iff $\int f(x) \mu_n(dx) \rightarrow \int f(x) \mu(dx)$ for all $f \in ...
2
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36 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
2
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135 views

How is the interchange of the limit and the maximum valid at this point in Erwin Kreyszig?

In 1.5-5 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, the author shows completeness of the space $C[a,b]$ of all (real- or complex-valued) functions defined and continuous ...
2
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58 views

Fourier transform of an exponentially singular radial function

I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int ...
2
votes
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84 views

Clarification: Rudin Theorem 3.7: Subsequential limits are closed

My question is this: Why does Rudin use $\delta$ in this proof? Would it not work just as well if $\forall i \ge1,$ $$x_{i}\in N_{2^{-(i+1)}}(q) \cap E^* $$ $$p_{n_i}\in N_{2^{-(i+1)}}(x_i) ...