For questions concerning the definition and properties of limit superior and limit inferior.

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16 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
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4answers
61 views

Superior limit of a certain sequence

Let $(x_n)$ be the sequence: $$\{1, 2, 1 + \frac12, 2 + \frac12, 1 + \frac13, 2 + \frac13, \ldots \}$$ I (think I) understand why $\lim \inf x_n = 1$, but I'm not sure why $\lim \sup x_n =2$. Any ...
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2answers
32 views

What is the difference of the greatest of the limits $\overline{\lim}_{n\to \infty}$ and the least of the limits $\underline{\lim}_{n\to \infty}$?? [on hold]

What are exactly these three limits for an infinite series $x_n$? $$\overline{\lim_{n\to \infty}} x_n$$ $$\underline{\lim}_{n\to \infty} x_n$$ $$\lim_{n\to \infty} x_n$$ Can they be different from ...
3
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0answers
35 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
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1answer
39 views

$\limsup = \liminf$ of sequence of Sets

This problem was on my in-class final for a measure theory course I took in the fall, and now I am studying for my qualifying exam so I am trying to figure this one out: Suppose ...
2
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1answer
47 views

Lim sup/inf of average value

Consider $$f(t)= \frac{1}{t} \int_{0}^t \sin(e^s) ds.$$ What is $$\mathrm{lim \ inf}_{t \rightarrow \infty} f(t)$$ and $$\mathrm{lim \ sup}_{t \rightarrow \infty} f(t)?$$ Using $u$-substitution, ...
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1answer
33 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
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2answers
59 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
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1answer
27 views

Applying Gronwall's Inequality

I've been working on this problem and was able to get it down to this inequality: $$E[\log X_t] \leq ae^{-t} + b(1-e^{-t}) + e^{-t}\int_0^t e^sE[\log X_s] ds$$ I'd like to use Gronwall's Inequality ...
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1answer
76 views

The value of limsup and liminf of a sequence of a sets obtained by combining three sequences

What is the limit superior of the following sequence of sets? $\{X_n\}=\{\{1/2\},\{1/3\},\{1/4\},\{2/3\},\{1/3\},\{1/5\},\{3/4\},\{1/3\},\{1/6\}......\}(n\to∞)$ I.e., $X_1=\{1/2\}, X_2=\{1/3\}, ...
3
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1answer
42 views

“Counterexample” to this characterization of lim sup?

I came across an exercise (Exercise 10, Ch. 1, Marsden's elementary classical analysis, 2nd ed.) that gives a characterization of lim sup I had never seen, which can be rephrased as follows: Let ...
2
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1answer
31 views

Are these implications true for a nonnegative stochastic process $X_t$?

Suppose I have a nonnegative stochastic process $X_t$. Furthermore, suppose the following is true: $$\limsup_t \frac{1}{t}E\left[ \log X(t)\right] \leq a < 0$$ for some constant $a \in ...
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0answers
16 views

Physical meaning about $\limsup$ of a Stochastic Process

I'm trying to show that a positive $n$-dimensional stochastic process $X_t = (x_1(t), \cdots, x_n(t))$ is nice in that it's well-behaved and controlled (in the sense that the process doesn't grow too ...
2
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1answer
52 views

Show That $\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$

My question concerns proving an inequality between two extreme limits, namely: $$\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$$ Using the following defintions: Let $f: E \to \mathbb{R}$ be a ...
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0answers
19 views

Is the limit superior defined on every real-valued function with a finite range?

I know the limit superior isn't always defined on function in general, but intuitively, I don't see how it could be undefined on a function with a finite range. However, I didn't find any sources ...
2
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0answers
18 views

$\limsup$ of Brownian Motion Time Integral

The following are well-known: $\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt ...
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0answers
53 views

What is bigger, $\liminf $ or $\inf ( \liminf )$

Let $E$ be a linear normed vector space, $f$ a functional $f : E \rightarrow \mathbb{R} $ and $ S = \{g \in E : \|g\| = 1\} \subset E$. Fix $x \in E$. I need to prove that $$\liminf_{g \in S, \ ...
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1answer
53 views

proof that $\limsup a_n=\sup\{a_n,a_{n+1},…\}$

How can I prove that $\limsup a_n= \sup\{a_n,a_{n+1},...\}$? I also need to prove: for two sequences $a_n>0$ and $b_n \ge 0$, then $\limsup(a_n b_n) \le \limsup(a_n) \limsup(b_n)$. I thought ...
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3answers
36 views

Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a normed space and $(x_n)_{n\in\mathbb{N}}\subseteq X$. Can we prove that there is a subsequence ...
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0answers
14 views

swap limsup with a function

What are the conditions that must hold for a sequence of real numbers $\{a_n\}$ and a real valued function $f$ so that the following relation holds: $$\limsup_{n\rightarrow\infty} f(a_n)= f( ...
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1answer
26 views

(analysis) sequence a(n) is bounbed and sequence b(n) converges. Show that a(n)≤b(n) (∀ n ∈ ℕ) ⇒lim supa(n)≤lim b(n) [closed]

sequence a(n) is bounbed and sequence b(n) converges. Show that a(n)≤b(n) (∀ n ∈ ℕ) ⇒lim sup a(n)≤lim b(n) Since a(n) is bounded, a(n) has a convergent subsequence. let a1'(n) be a subsequence of ...
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2answers
38 views

Let $ f:I \to \mathbb{R} , I=(0,1) $ be uniformly continuous. Then exists $ \lim_{n\to\infty} f(\frac{1}{n}) $

True. Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume: $$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$ Since $ \lim_{n\to\infty} \frac{1}{n} $ is ...
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1answer
28 views

Find $\liminf\limits_{n\to\infty} (x_{n})$ and $\limsup\limits_{n\to\infty}(x_{n})$ for a sequence $x_{n}=1-nsin\frac{n\pi}{4}$

Subsequence $a_{n_{1}}=sin\frac{n\pi}{4}$ is bounded ($[-1,1]$), and a subsequence $a_{n_{2}}=n$ is bounded below. We can find cluster points for the first subsequence $C_{1}=\{-1,1\}$. For the second ...
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1answer
24 views

My attempt at finding $\underline{\lim_{n \to \infty}}A_n$ and $\overline{\lim_{n \to \infty}}A_n$ $A_n=B$ for $n$ even; $A_n=C$ for $n$ odd.

My attempt at finding $$a.) \underline{\lim_{n \to \infty}}A_n$$ and $$b.) \overline{\lim_{n \to \infty}}A_n$$ $A_n=B$ for $n$ even; $A_n=C$ for $n$ odd. $a.)=\bigcup_{n=1}^{\infty} ...
0
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1answer
47 views

Asymptotic sums and liminf

Given an arithmetic function $f(n)>0$ with $$\liminf \frac{g(n)}{f(n)}=C$$ for a certain constant $C$ and another function $g(n)>0$, in the study of the asymptotic bound for $$ \sum_{n\leq x} ...
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2answers
29 views

If $a_n \geq b_n$, $b_n \leq |c_n|$ with $c_n \to 0$, does this imply $\liminf a_n \geq 0?$

Let $a_n \geq b_n$ where $b_n \leq |c_n|$ with $c_n \to 0$. Does this imply that $$\liminf a_n \geq 0?$$ These are all real-valued sequences. I don't think it is enough to conclude.
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0answers
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When working with the limites inferior and superior, how do you procede?

There are typical cases of various complexity where notions are defined or solvable by liminf or limsup. At the same time, from my perception, they are touched only superficially during the course of ...
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2answers
32 views

Yes or No: The following $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ holds

Can someone verify whether $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ is mathematically rigorous?
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1answer
11 views

Limit superior and inferior of a sequence that satisfies the asymptotic formula $\sum\limits_{n\leq x} a_n \sim x $

Suppose $\{a_n\}$ is a sequence such that $\sum\limits_{n\leq x} a_n \sim x $. I have to show that: $$ \liminf a_n\leq 1 \leq \limsup a_n$$ I've no idea on how to approach this, in all honesty.
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0answers
30 views

Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
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1answer
47 views

Prove a limsup and liminf inequality.

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence and let $B_n = \frac{1}{n} \sum_{i=1}^n a_i$ for each $n \in \mathbb{N}$. Prove that $\liminf a_n \le \liminf B_n \le \limsup B_n \le \limsup a_n$. ...
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0answers
34 views

References for limit superior and limit inferior of functions

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{\pm \infty\}$. I would like to find books for the following notions and their properties: 1) $\limsup_{x\rightarrow x_0}f(x), \liminf_{x\rightarrow ...
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1answer
17 views

Limit inferior implication

I'm having trouble with the following statement. If $M=\liminf_{t\uparrow b}|x(t)|$ then $\exists$ a sequence $t_n\rightarrow b$ such that |$x(t_n)| \leq M+1$ I take it that with the sequence ...
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2answers
160 views

Computing a limit of sequence

After applying l'Hopital rule twice, one sees that $$ \lim_{n\to \infty} n a ~e^{-an} =0 \qquad \qquad (a\in [0,1]) . $$ I would like to ask if someone can prove it using different way? Bests.
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2answers
31 views

Is $\limsup_{z\to z_0}f(z)=\limsup_{k\to\infty}f(z_k)$?

Let $\Omega\in\Bbb C$ open, $f:\Omega\to\Bbb R$ a generic function. Let $(z_k)_k\subset\Omega$ s.t. $\lim_{k}z_k=:z_0\in\Omega$. The question is the following: is true that $$ \limsup_{z\to ...
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1answer
33 views

Proof of Fatou-Lebesgue Theorem

Good evening everyone, how can I prove the following inequality? Let $f,f_n\in L(X,\mu)$ on a measure space with $0\leq f_n(x)\leq f(x)$. If $f,f_n$ are $\mu$-almost everywhere in $X$, then is ...
0
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1answer
35 views

Pointwise convergence of $c$-concave envelopes as the sequence of cost functions converges pointwise

On Remark 1.12, page 33 of Villani's Topics in optimal transport we find the following problem. It should be easy but it's turning me mad. Let $c:\mathbb{R}^n\times\mathbb{R}^n\rightarrow ...
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3answers
29 views

Finding two sequences with a limsup value

Find two sequences ⟨ X n ⟩ and ⟨ Y n ⟩ such that limsup(Xn + Yn )=E=Euler's constant and limsup Xn + limsup Yn =Pi=π . I couldn't ...
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0answers
18 views

Liminf Brownian Motion question

For this assignment I'm working on, I was able to prove that: $$\limsup_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = \infty$$ where $B_t$ is a Brownian Motion. I'd like to be able to prove: ...
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2answers
49 views

Isn't the statement of the Fatou's lemma somewhat problematic?

My lecture notes define $\int f := \int f^+ - \int f^-$ provided both $\int f^{\pm}$ are finite. And then the Fatou's lemma is stated in the following way: Let $f_n$ be a sequence of integrable ...
4
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1answer
56 views

If $\limsup_n\sqrt[n]{a_n}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{(n+1)a_{n+1}}=?$

If $\limsup_n\sqrt[n]{|a_n|}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{|(n+1)a_{n+1}|}=?$ Can I separate the product of the second limit as ...
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3answers
64 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .

Note: $x_n$ is a sequence which is not necessarily convergent. The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also ...
4
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0answers
37 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
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2answers
71 views

Excerise 12.2 from Ross Elementary Analysis

I'm having a little bit of difficulty proving this question: Prove that $ \limsup_{n->\infty} |S_n| = 0 ~~$iff$ ~\lim_{n->\infty}S_n = 0. $ What I have so far: $ (\Leftarrow)$ $ $Suppose $ ...
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1answer
30 views

Every sequence of sets have coverging subsequence

Is that true, that every sequence of sets have coverging subsequence? We say that sequence of sets $A_1, A_2, A_3, ...$ coverging iff ${\limsup}_{n \to \infty} A_n = {\liminf}_{n \to \infty} A_n$
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1answer
36 views

What is the limsup of the following sequence of sets?

Problem: Let $A_n=\{\frac{m}{n}:m\in\mathbb{Z}\}$. Find $\limsup_{n\rightarrow\infty}A_n$ and $\liminf_{n\rightarrow\infty}A_n$ Attempted: It is clear that limsup should be $\mathbb{Q}$. I can show ...
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1answer
50 views

Give an example to show that the inequalities are strict inequalities

Give an example to show that the following three inequalities $$\liminf_{n \to \infty} (a_n) +\liminf_{n \to \infty} (b_n)\le\liminf_{n \to \infty} (a_n+b_n)\le\limsup_{n \to \infty} (a_n+b_n) \le ...
0
votes
1answer
45 views

Doubt on Tail events and Kolmogorov Zero-One Law

In this wiki article on the law of the iterated logarithm, one states that, given $M>0$, the event $$A=\left\{\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}} > M\right\}$$ has probability $0$ or ...
2
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0answers
52 views

how to show $\lim \sup_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= \infty$ and $\lim \inf_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= 0$?

Let $\phi (n)$ be the Euler's totient function . Thenhow to prove that the set $\{\dfrac{\phi(n+1)}{\phi(n)}: n \in \mathbb Z^+\}$ is unbounded above that is how to show $\lim \sup_{n \to \infty} ...
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1answer
36 views

Prove that $\limsup_{n \to \infty} (a_n+b_n)= a + \limsup_{n \to \infty} b_n$

I have to prove the following: Given that lim $a_n$ exists and that lim $a_n$ = $a\in \mathbb{R}$ prove that: $$\limsup_{n \to \infty} (a_n+b_n)= a + \limsup_{n \to \infty} b_n$$ I proved one ...