Convergence of sequences and different modes of convergence.
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28 views
convergence of discrete random variables with finite entropy
Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
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55 views
sequence with almost surely convergence & moment of order 2
Let's say we have iid random variables $(X_n)$ s.t. $X_1$ admits a moment of order 2.
For $n \geq 1$, does the following sequence converge almost surely or not? Why? And how to see/show?
$$A_n = ...
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43 views
Taylor series of Fourier series of triangle wave
Odd triangle wave $\text{t}(x)$ with angles at $(2x+1)\in\mathbb{Z}$ can be represented by Fourier series:
...
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0answers
60 views
Proof of convergence
Prove if the following series converges:
1) $\sum_{k=2}^\infty \frac{k^2}{-1+k^5}$
$$\lim\limits_{n \to \infty}\frac{k^2}{-1+k^5} \cdot\frac{1}{1/k^3}$$
$$\Leftrightarrow \lim\limits_{n \to ...
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0answers
42 views
Time steps and convergence
I was wondering if someone could please give me an explanation why we look at larger and larger time steps in order to see if we have convergence. I have played around with matlab and noticed when I ...
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51 views
Dense subset of the conjugate space
The question is:
Let $X$ be a normed linear space and let $B$ be a dense subset of $X^*$(the conjugate space of $X$). If a sequence $\{x_n\}$ in $X$ is bounded, and if $\lim_n x^*(x_n)$ exists for ...
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0answers
65 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 ...
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155 views
Rademacher function and weak convergence
The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
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71 views
Uniform convergence in $L^p$-spaces
Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$.
...
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78 views
what is sub-linear convergence called?
What is sub-linear reciprocal convergence called? Would it be called parabolic convergence? For example, consider the series:
...
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42 views
How to argue this “standard elliptic estimates” for a convergence?
I am estudying the paper: INEQUALITIES FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH APPLICATIONS TO UNBOUNDED
DOMAINS I - H. BERESTYCKI, L. A. CAFFARELLI, AND L. NIRENBERG
At the page $482$, we have a ...
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42 views
Convergence of integrals over divergent parts
I'm wondering if it is possible for an integral which diverges in the limits $1$ to $\infty$ to converge in the limits from $0$ to $\infty$. And if so: how could I find this out?
For example $$ ...
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0answers
96 views
Convergence of the reciprocal of a function whose derivative tends to infinity
If $$\lim_{x\to\infty} f'(x) = \infty$$ prove that $$\int_1^\infty \frac{1}{f(x)}\neq\infty$$
if $f'(x) \geq 1$ and $f(x) \geq1$ for all values of $x$. I'm thinking I can find a way to write it in big ...
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37 views
Carry convergence from one series to another
Suppose I have
(i) $\alpha_n \geq 0$, $\beta_n \geq 0$ such that
(ii) $\sum_{n=1}^\infty \alpha_n = \sum_{n=1}^\infty \beta_n = 1$, and
(iii) $\sum_{n=1}^\infty n\alpha_n < \infty$. ...
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0answers
22 views
Convergence of a serie depending on an exponent
For which $k>0$ is $\sum \limits_{n=1}^\infty (\frac {1}{n+n^k})$ convergent.
I tried to prove convergence with the Ration test but it hasn't really worked.
by the P-test and comparison test I ...
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0answers
149 views
different kind of convergence in Real analysis
Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
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77 views
Example of compactness of the set of elements of a convergent sequence.
An example in a book I am reading on general topology demonstrates the concept of compactness by showing that the set $E = \{s_n : n = 0,1,2,3 \ldots \}$ is compact in some topological space S. The ...
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0answers
31 views
Rate of convergence of a single-neuron Perceptron network
I'm implementing a Perceptron network which basically consists of a single neuron in a single layer, trying to learn an OR logic port (linearly separable), but using the sigmoid function as ...
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0answers
55 views
Uniform convergence of complex exponent derivative
I'm trying to prove the following:
Let $\Re z > 0$. Then $$\lim_{\varepsilon \to 0} \frac{t^{z + \varepsilon} - t^z}{\varepsilon} = t^z \log t$$ uniformly in $t \in [0,1]$.
I've tried to ...
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67 views
convergence uniformly of a sequence of functions
Show that $$f_n: \mathbb{R} \rightarrow \mathbb{R}; \ n\ge1; \ f_n (x)=\frac{x \sqrt{n}}{n \sqrt{n}+x^2}$$ does not converge uniformly on $\mathbb{R}$
I have showed that $f_n(x)$ pointwise converges ...
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33 views
The sum of an unconditionally summable sequence does not depend on the order of the terms
If I have a series of elements of a Banach space that converges unconditionally (it converges regardless of the order of the terms), why is it the case that the sum does not depend on the order of the ...
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111 views
If $x_n$ and $y_n$ are Cauchy and $y_n \neq 0$ for all $n \in \mathbb{N}$, is ${\frac{x_n}{y_n}}$ also Cauchy?
My current proof:
As $x_n$ and $y_n$ are Cauchy, they are both convergent.
Then, as $y_n \neq 0$ for all $n \in \mathbb{N}$, then ${\frac{x_n}{y_n}}$ is also convergent.
Thus, as $\frac{x_n}{y_n}$ is ...
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54 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 ...
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0answers
62 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 ...
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0answers
48 views
Estimating the number of observations from a set of samples
I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs.
All of the hidden inputs are driven by an experimenter ...
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0answers
47 views
Rationality and convergences
I observed that convergence of partial fractions and I am writing this samll story for seeking more clarity and justifications etc.
If f(n) = $\frac{p(n)}{q(n)}$, where p and q are polynomials of n ...
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0answers
72 views
Convolution with a particular delta sequence
Let $f:\mathbb{R}\to\mathbb{R}$ be locally integrable and define a function $\rho_L$ for $L>0$ by $$\rho_L(x) := \frac1{x\ln L} 1_{[1/L,1]}(x).$$
Obviously $\int\rho_L = 1$. Is it true that the ...
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81 views
Does higher convergence order guarantee higher convergence speed?
We say that the sequence converges with order $q$ to $L$ for $q>1$ if
$$ \lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = \mu, \quad\; \mu > 0.$$
If $a_k$ converges with order $p$, $b_k$ ...
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0answers
126 views
Limit alternating series
Show that for any $b>0$ and $n>0$, the alternating sum of the series
$$
a_k=\prod_{i=1}^k\frac{1+b/(n+i)}{1+b/(n+1)}
$$
converges such that
$$
\sum_{k=1}^{\infty} (-1)^k a_k \leq -\frac{1}{2}
...
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0answers
482 views
Simple example application of Karush-Kuhn-Tucker conditions to minimization problem
I am wondering if there is a simple example application of the Karush-Kuhn-Tucker conditions to show that a minimum exists for a multivariate minimization/optimization problem. Could anyone suggest a ...
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0answers
93 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 ...
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0answers
112 views
Constructing Ito integral for adapted process
I am trying to construct Ito integral for adapted process. However, I am stuck at some point.
Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the ...
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0answers
252 views
Convergence of sequence of analytic functions
Given $\{f_{n}(z)\}$, a sequence of analytic functions in the upper half plane $\mathbb C^{+}$, where each $f_{n}(z)$ has continuous extension to the real line, and $|f_{n}(z)|\leq 1$ for all $z\in ...
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0answers
249 views
Convergence of $L^p$ norms
Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that
$\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, ...
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0answers
261 views
showing that a sequence is uniformly integrable
I am currently reading the new edition of Royden and I've gotten to a part where the book made some comments without justification and I'm trying to verify these facts on my own. I want your help in ...
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0answers
345 views
So is this, finally, the difference between convergence in probability and almost sure convergence?
I've been trying to come up with a intuitive, practical distinction between convergence in probability and convergence almost surely. Can someone please tell me if the following is correct?
Let $X_n$ ...
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0answers
159 views
Convergence of the Taylor series for the sine function
I would like to know if the Taylor series for the sine function,
$$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots,$$
is convergent if the argument of the function, $x$, is expressed in ...
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0answers
292 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 ...
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0answers
168 views
Applications of Convergence of a series in Algorithms
We were introduced to testing the convergence of a series & calculating the point of convergence in the first maths of college curriculum. I wish to explore its usage in computer algorithms.
...
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0answers
166 views
Convergent in $L^1(0,1)$ but not in $L^2(0,1)$ help understanding a paper from arxiv
http://arxiv.org/pdf/math/0205003v1
In around equation (1.1) the author says
"By necessity all authors have been led in one way or another to the natural approximation
$$F(n) := \sum_{a=1}^n \mu(a) ...
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0answers
18 views
Convergence of density function
Let $X_m$ have the density function
$$
f_m(x) = \frac{m}{ \pi(1+m^2x^2)}
$$
where $m \ge 1$. Which modes of convergence have to be respected that $X_m$ converges (if $n \rightarrow \infty$) ?
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0answers
31 views
Does $f(X_n)\to f(X)$ in probability imply $X_n\to X$ in probability?
Does $f(X_n)$ converge in probability to $f(X)$ imply $X_n$ converge in probability to $X$?
0
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0answers
97 views
Qualities of Projected Gradient Methods
Consider the following constrained minimization problem:
$ min_{x \in X} \ f(x) $
where $ X \subset \Bbb{R}^{n} $ is a nonempty closed convex set and f is continuously diferentiable.
I'm ...
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0answers
26 views
show $\frac{1}{n}\sum (X_i - \mu_i) \xrightarrow{L^2} 0$, $X_i$ independent with mean $\mu_i$
Let $\{X_i\}_{i=1}^n$ be a sequence of uncorrelated random variables,
$$
\text{Cov}(X_i,X_j)=0,
$$
with
$$
E(X_i)=\mu_i<\infty,
$$
and
$$
\frac{V(X_i)}{i} \xrightarrow{i\to \infty} 0. \tag{1}
...
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0answers
9 views
How can the distribution characteristics in different random sample sizes be known?
Here is my logic:
A sample from a random pool (algorithm or natural) is known to have a certain distribution (normal, uniform, etc.) assuming both the sample and the pool (population?) are big ...
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0answers
12 views
Weak-Sense Convergence
I am looking for online resources about weak-sense convergence for eulers scheme. Note sure if question is appropriately placed but was hoping for some recommendations?
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0answers
19 views
Monte Carlo Technique and Convergence
I am solving $I = \int_{0}^{1}f(x)dx$ by Monte Carlo, e.g $I = E[f(U)]$ where $U\sim unif([0,1])$ so have
$I = E[f(U)]\approx \hat I_M\colon= \frac{1}{M}\sum_{m=1}^Mf(u_m)$
I am interested in how ...
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0answers
16 views
Second Moments: Convergence in Probability
I wish to confirm my answer to the following question:
Let X1, X2,…., Xn be iid random variables with mean µ and variance $\sigma ^{2}$ and Var($X_{1}^{2}$) = k < ∞. Show that:
...
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0answers
31 views
a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?
Any example that a sequence of random variables that converge to a
constant c in probability but fail to converge to c with probability 1?
0
votes
0answers
25 views
On the absolute convergence of some function.
Let $f(s,g)$ be a two variable smooth good (in a suitable sense) function and let $F(s)=\int_{G}f(s,g)dg$. Assume $F(s)$ is absolutely convergent for $\Re(s)>0$ and has meromorphic continuation to ...
