Questions tagged [tauberian-theory]
Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result.
33
questions
1
vote
0
answers
54
views
Informations about a sequence from tail behaviour
Suppose $\{c_n\}_n$ is a sequence of non negative reals. We have the following three informations about it.
(a) $\sum_{k \ge n}c_k \sim \frac{1}{2n}$
(b) $\sum_{k=2}^n \frac{kc_k}{\log n} \to \frac{1}{...
- 765
2
votes
0
answers
83
views
Tauberian theorems for Laplace transform
$X_1$ ($(X_n)_{n\geq 1}$ is a skip free random walk, i.e. $(X_n)_{n\geq 1}$ i.i.d takes value in {-1,0,1,2,...} with $\mathbb{E}(X_n)=0$ for all $n\geq 1$ and $\mathbb{P}(X_1+1=k)\sim Ck^{-\alpha-1}$ ...
user1118550
0
votes
1
answer
61
views
A converge problem seems to related to Tauber theorem
ps: Due to my poor English, I might describe my thought roughly.
Suppose $\{a_n\} (n \ge 0)$ is a sequence consisting of non-negative numbers,
and $\{a_n\}$ satisfies that forall $x > 0$, $f(x) = \...
2
votes
1
answer
44
views
Negativity of an analytic function on $[0,1)$
I want to show the following function is negative for $z\in [0,1)$:
$$f(z) = -1 + z^2(z-1) + 2\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2+1}. $$
By Tauberian theorem, I know that $\lim_{z\to 1^-}f(z)=0$. I ...
- 389
2
votes
0
answers
83
views
Asymptotics of moment generating function
Consider r.v. $\xi$ with known c.d.f. $F$ and p.d.f. $f$. Let the corresponding moment generating function $M(z)$ be finite for all $z \in \mathbb{R}$. I am interested in deriving the asymptotics of $...
- 129
2
votes
0
answers
156
views
Hardy–Littlewood Tauberian theorem for Laplace transform
The Hardy–Littlewood Tauberian theorem for Laplace transform in Chapter XIII in "An Introduction to Probability Theory and Its Applications" by Feller reads as follows
Let $F : [0,\infty) \...
- 413
20
votes
5
answers
786
views
Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$
This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit
$$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$
It is clear that the ...
- 935
1
vote
0
answers
43
views
Convergence speed of the tail of distribution using Tauberian remainder theorem
This question is related to this.
Now I try to make some statistical inference using Laplace transform, but I face the following problem.
Let $f$ be some one-sided probability distribution defined on $...
- 715
2
votes
1
answer
193
views
How to prove this Tauberian theorem
How to prove this exercise on tauberian theorem from Zorich:
Tauber's original theorem relates to Abel summation of series and consists
of the following.
Suppose the series $\sum\limits_{n=1}^\infty ...
- 783
4
votes
1
answer
89
views
Limiting behavior of a function defined by a Lambert-type series
Fix a positive real constant $\omega$, and let $\left\{ c_{n}\right\} _{n\geq1}$ be a sequence of real numbers so that the series: $$\sum_{n=1}^{\infty}\frac{c_{n}}{n^{\omega}}$$ converges ...
- 2,219
2
votes
0
answers
98
views
Asymptotic expansion at $x=1^-$ for $\sum_{n=1}^{\infty} x^{a_n}$
I feel that this general fact should be known. Suppose I have a strictly increasing sequence of integer positive numbers $\{a_n\}_{n \in \mathbb{N}}$. We want to investigate the behavior of
$$ f(x)= \...
- 193
1
vote
0
answers
69
views
From averages with weight $1/x \log e x$ to averages with weight $1/x$
Let $S:[1,\infty)\to \mathbb{R}$ be a function with $0\leq S(x)\leq 1$ for all $x$. How do I go from estimates on integrals of the form
$$\int_1^w S(x)\; d \log \log e x$$to estimates on integrals of ...
- 1,324
1
vote
1
answer
40
views
Asymptotics of convolution of a series given the asymptotics of the series
so I have a series of real numbers $(q_n)_{n\in\mathbb N}$ ($q_i \in [0,1]$) depending on some $\alpha \in (0,1/2)$ and what I know is the following, there exists constants $c_1, c_2$ so that:
$q_i \...
- 317
0
votes
1
answer
131
views
Why is this not a counter-example of the Hardy-Littlewood tauberian theorem?
I am confused about the Hardy-Littlewood tauberian theorem. If we apply it with the sequence $a_n$ whose first few terms are given by $1,-1,1,1,-1,-1,1,1,1,1,-1,-1,-1,-1...$ where we put $2^n$ $1$s ...
- 2,527
1
vote
0
answers
122
views
Weak Tauberian theorem
Karamata's Tauberian theorem states the following. Let $A(z)=\sum a_nz^n$ be a power series with non-negative coefficients $a_n$ and radius of convergence 1. Then, $\sum_{n\geq 0}s^n\underset{s\to 1}{\...
- 299
1
vote
0
answers
48
views
Hardy-Littlewood Tauberian Theorem for stochastic processes
For which processes do we have
$$
\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T} X_t dt =\lim_{a\to 0}a\int_{0}^{\infty} e^{-at}X_t dt
$$
almost surely?
The Hardy Littlewood implies that this holds for ...
- 8,008
-1
votes
1
answer
64
views
A Converse For A Particular Case of the Hardy-Littlewood Tauberian Theorem
Let $V$ be a set of positive integers, and let: $$\varsigma_{V}\left(x\right)\overset{\textrm{def}}{=}\sum_{v\in V}x^{v}$$ Defining the natural density of $V$ by the limit: $$d\left(V\right)\overset{\...
- 2,219
3
votes
0
answers
169
views
Karamata's proof of Hardy-Littlewood Tauberian theorem
I understand Karamata's proof of Hardy Littlewood Tauberian theorem here, but what on earth is the motivation behind Lemma 4 - i.e. what would be the motivation to look at the space of all functions $...
- 1,872
1
vote
0
answers
125
views
Remainder in the Wiener-Ikehara theorem
I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form
$$ \sum_{n\leq X} a(n) = cX^a(\log X)^m + O\big(X(\log X)^{m-1}\big),$$
i.e. ...
- 661
1
vote
0
answers
76
views
Typo in first Tauberian theorem in Wiener's "Tauberian Theorems"?
In Equation (0.0.7)-(0.09) of Norbert Wiener's Tauberian Theorems, it is stated that
[...] if $$ \lim_{x\to 1-0}\frac{1}{1-x}\sum_{0}^{\infty}a_nx^n=A, $$ and
$$ a_n= o(1/n), $$ then $$ \sum_{0}...
- 8,008
0
votes
1
answer
182
views
Hardy's power series. Cesàro convergence
Doing my analysis homework i have come across the following power series known as Hardy's power series
$$\sum\limits_{k=0}^{\infty}a_kx^k=\sum\limits_{k=0}^{\infty}(-1)^kx^{2^k}\mbox{ for x}\in[0,1],$$...
- 114
4
votes
1
answer
164
views
A Tauberian theorem for a quotient of power series, the limit on the boundary
Take sequences $a_n, b_n \in \mathbb R_{>0}$ (or $\mathbb C$) such that the limit $$L = \lim_{N \to \infty}\frac{\sum_{n \leq N} a_n }{\sum_{n \leq N} b_n}$$
exists and such that the power series ...
- 26.6k
2
votes
0
answers
405
views
Laplace transform and tail probability
Let $X \ge 0$ be a non-negative random variable. I would like to know if the following statements are equivalent:
$$ \lim_{\lambda \to 0^+} \frac{\mathbb{E} \left[X e^{-\lambda X}\right]}{\log (1/\...
- 343
5
votes
0
answers
503
views
Fourier series: $\hat f(n)=O(1/n)$ and $f$ continuous implies uniform convergence?
Littlewood's Tauberian theorem: Let $a_n=O(1/n)$. (In particular, given any $0<r<1$, the power series $\sum a_nr^n$ converges.) If the function defined by the power series $$f(r)=\sum a_nr^n$$ ...
- 2,778
0
votes
1
answer
200
views
Final Value Theorem Variant
Let $f:\mathbb{R}\to\mathbb{R}$ be given (possibly with some conditions to be added later?).
Prove the following statement:
$$ \lim_{x\to\infty}\frac{1}{x}\int_0^xdy\int_0^y\,dz\,f(z)=\lim_{s\to0}\...
- 1,116
2
votes
2
answers
210
views
A couple of difficulties in Tauber theory lecture notes of prof. Yum-Tong Siu
Reading trough the lecture notes on Tauber theory of prof. Yum-Tong Siu I am a bit off right at the beginning. If someone could clarify the follow two steps in his proofs of Tauber's original 1897 ...
- 457
3
votes
1
answer
755
views
Verifying $|F(r)| \geq \frac{1}{1-r}\log(\frac{1}{1-r}) $ and $|F(re^{i \theta})| \geq c_{q/r}\frac{1}{1-r}\log({\log(\frac{1}{1-r})})$
I'm attempting to take a Tauberian route in verifying the proposition in $(1)$ below, which is from Complex Analysis, by Elias M Stein and Rami M. Shakarchi.
Let $F(z)$ be the following series:
$$F(...
- 1,071
0
votes
0
answers
659
views
Showing Abel Means of a Fourier Series Converge Uniformaly to $f$?
In the text "Fourier Analysis and Related Topics", i'm having trouble proving the following Theorem in $(3.5.5)$ utilizing Fourier Methods/Summability Methods. Also i'm not sure how to approach $(ii)$ ...
- 1,071
0
votes
2
answers
1k
views
Proving $\sin(x) + \sin(2x) + \sin(3x)+ \cdots +\sin(nx) = \frac{cos(1/2) - cos(n-1/2)}{2 \sin(1/2)2}$ via Ceasro Summation?
I'm having trouble proving the following conjecture in $(1.)$, via Fourier methods, my intial attack can be seen in $(2)$
$(1.)$
$$\sin(x) + \sin(2x) + \sin(3x)+ \cdots +\sin(nx) = \frac{cos(1/2) - ...
- 1,071
1
vote
0
answers
291
views
Showing the Series $\sum_{n=1}^{\infty}c_{n}$ convergences and that it's abel summable to s?
In stein's Fourier Analysis text i'm attempting to verifying my proof that the $\sum_{n=1}^{\infty}c_{n}$ converges to a finite limit $s$ then the series is Abel summable to $s$. My initial attempt ...
- 1,071
1
vote
0
answers
35
views
Uniform tauberian theorem
Let $\epsilon>0$.
I wonder if the following is true:
There exists $\lambda_0\in(0,1)$, such that for all $\lambda\leq\lambda_0$ and all $(a_n)\in [0,1]^\mathbb{N}$ there exists $N\in \mathbb{N}$ ...
5
votes
0
answers
669
views
Differentiating an asymptotic power series
I am interested in theorems that allow me to differentiate a divergent asymptotic power series for $x \to \infty$. I have a function $f:\mathbb{R}\to\mathbb{R}$ that is differentiable for large enough ...
- 825
4
votes
0
answers
199
views
Example and counter-examples for the tauberian theorem of Hardy-Littlewood
We know the following theorem:
Theorem (Hardy-Littlewood). Let $(b_n)$ be a real sequence such that
(i) $(nb_n)$ is bounded,
(ii) $\displaystyle\lim_{x\to 1^-} \sum_{n=0}^\infty b_n x^n=\ell$.
Then $\...
- 14.9k