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

learn more… | top users | synonyms (2)

0
votes
0answers
21 views

relations between the root test and the ratio test

relations between the root test and the ratio test I know the theorem is correct if they are exist $$ \lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow ...
1
vote
0answers
34 views

Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
0
votes
0answers
12 views

radius of symmetric random walk on $\mathbb{Z}$

How to calculate the radius of the symmetric random walk on $\mathbb{Z}$, i.e. $\limsup_k (p^{(k)}(0,0))^\frac{1}{k}$? ($p^{(k)}(0,0)$ denotes the probability to get from $0$ to $0$ in $k$ steps and ...
0
votes
1answer
25 views

How to show that the limit of a function does not exist, while $\limsup$ and $\liminf$ do?

It does seem like a very simple question, but I don't see how to do it properly. How can we show that the limit of $$3^{\lfloor\frac{-\log(j)}{\log(2)}\rfloor}j^{\frac{\log(3)}{\log(2)}}$$ as ...
1
vote
3answers
37 views

Lim sup definition doesn't make sense to me

I don't really understand the concept of Lim Sup and why it is any different from a standard limit. I am working on a question about radius of convergence and I know that $r = ...
0
votes
1answer
22 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
1
vote
1answer
61 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
0
votes
0answers
28 views

What are the lim sups and lim infs of a sequence?

What relation, if any, can you state for the lim sups and lim infs of a sequence {an} and one of its subsequences {$a_n$$_k$}?
7
votes
5answers
364 views

Understanding limsup

My textbook says: $$\overline{s}_n = \sup \{a_n \mid n \geq N\}$$ and $\operatorname{limsup} \{a_n\}_{n \to \infty} = \lim_{N \to \infty} \overline{s}_N$. Also, it says: As $N$ gets larger, the sup ...
0
votes
0answers
21 views

Limit points and limsup

Suppose {$a_n$} is a bounded sequence, and P is the set of limit points. Prove that $\limsup a_n = \sup P$. Don't know what the steps are...
1
vote
2answers
47 views

$\limsup \sqrt[n]{a_n} \leq \limsup a_n$ if $(a_n)$ is non-negative sequence?

Let $(a_n)$ be a sequence with $0 \leq a_n \leq 1$. Is it possible to show, that $\limsup \sqrt[n]{a_n} \leq \limsup a_n$? Conversely, if $(a_n)$ is a sequence with $0 \leq a_n$ and $\limsup ...
0
votes
0answers
26 views

Stochastic big O notation

Let $||.||$ indicate the Euclidean norm. Let $\theta_0$ be a specific value of the parameter $\theta \in \Theta\subseteq \mathbb{R}^d$ and $G_n$ be a random vector-valued function $$ G_n : \Theta ...
0
votes
1answer
43 views

Help proving a limit exists using limsup and liminf.

Let $X=(x_n)_{n\in N}$ be a bounded sequence of real numbers with $\liminf_{n \to \infty}x_n=\limsup_{n \to \infty}x_n$. Prove that $X$ is convergent. I'm not sure where to start. Any suggestions?
2
votes
1answer
55 views

Proving the Kochen-Stone lemma using the Paley-Zygmund inequality

I am trying to understand a proof to a lemma by Kochen and Stone which appears here, using the Paley-Zygmund inequality. I'll repeat the proof in a detailed manner, and explain what bothers me about ...
0
votes
0answers
25 views

Oscillation, diameter and limsup

Let $I\subset \mathbb{R}$ be an interval. By definition $\text{diam}f(I)=\sup_{x, x'\in I}|f(x)-f(x')|$, $\text{osc}(f, I)=\sup_{x\in I}f(x)-\inf_{x\in I}f(x)$ and $\text{osc}(f, ...
0
votes
1answer
33 views

$\inf$ of a sequence of $\sup$

let $f_n$ a sequence of right-continuous functions $[0,\infty) \to R$. Assume that for each $T$: \begin{align} & \sup_{\substack{t \in [0,T]\\n \in N}} |f_n(t)|<\infty \end{align} Is it true ...
0
votes
1answer
19 views

Refining the asymptotics of a sequence

Assume that we have a monotonously increasing sequence $L_k$, such that $\lim_{k\rightarrow \infty}\left(L_kb^{-k}\right)=0$ and $\lim_{k\rightarrow \infty}\left(L_ka^{-k}\right)=\infty$, can we ...
0
votes
1answer
19 views

$\lim\sup(|s_n|) = 0$ iff $\lim(s_n) = 0$

I need to prove that $\lim\sup(|s_n|) = 0$ if and only if $\lim(s_n) = 0$. I totally get the intuition behind this implication but not sure how to give a formal proof of it. Here is my intuition: for ...
0
votes
1answer
27 views

Question regarding the formal definition of limes Inferior/Superior

I have a question regarding the definitions of limes inferior/superior of a sequence $x_n$. Limes inferior for example is defined as $$lim ~ sup_{n \rightarrow \infty} x_n := lim_{n \rightarrow ...
1
vote
2answers
36 views

$\lim \sup$ of a sequence

Let $\{A_n\}$ be a sequence and $\frac{1}{R} = \lim \sup A_n$. Let $\alpha < R$. My question: Why is there $n_0\in \Bbb N $ such that $$A_n < \frac{1}{\alpha}\text{ for any } n\geq n_0$$ Thanks ...
1
vote
0answers
15 views

limsup and quotient of sequences

Let $r\notin\mathbb{Q}$ and $\left\{ a_{n}\right\} _{n\in\mathbb{N}},\left\{ b_{n}\right\} _{n\in\mathbb{N}}$ such that $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=r$. Find $\limsup a_{n}$ and $\limsup ...
2
votes
1answer
43 views

Set convergence and lim inf and lim sup

I'm a bit confused with the general concept of convergence of a sequence of sets. I'm well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = ...
0
votes
1answer
53 views

To prove that $\limsup_{n\to\infty}\frac{1}{a_n} = \frac 1{\liminf_{n\to\infty}a_n}$

I want to prove this equality: $$\limsup_{n\to\infty}\frac{1}{a_n} = \frac{1}{\liminf_{n\to\infty}a_n},\; \forall n\ a_n > 0.$$ What I was thinking about was: Just following the definition, let ...
0
votes
1answer
65 views

$\lim\sup$ vs. $\lim$ as $n \to \infty$

What is the difference between the expressions $$ \lim\sup\left|\frac{a_{n+1}}{a_n}\right| \ \ \ \ \ \mbox{and} \ \ \ \ \ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| $$ specifically with ...
-1
votes
1answer
57 views

Show, using the definitions, that $\lim_{n \rightarrow \infty} \inf a_n = \infty$ implies that $\lim_{n \rightarrow \infty} a_n = \infty$

Show, using the definitions, that $\lim_{n \rightarrow \infty} \inf a_n = \infty$ implies that $\lim_{n \rightarrow \infty} a_n = \infty$ Here is the definition of $\liminf$ is $\liminf_{n ...
2
votes
1answer
24 views

What is the difference between limit inferior and limit?

So, I am working on problems on $L_p$ spaces. Every time I think of taking limits of both sides of an equation, the solution seems to take limit inferior/ limit superior instead and use the relation ...
2
votes
1answer
55 views

Show that $\lim \inf a_n\le\lim\inf s_n.$

Let $\{a_n\}$ be a bounded sequence of real numbers. Let $s_n=\dfrac{a_1+a_2+\cdots+a_n}{n}~\forall~n\in\mathbb N.$ Show that $\lim \inf a_n\le\lim\inf s_n.$ The only definition I know of limit ...
2
votes
0answers
59 views

Proof for $\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$

How I can show the truth of this relationship? $$\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$$ The definition of $\limsup x_n$ is $$\limsup x_n = \inf_n\left[\sup_{k\ge ...
0
votes
0answers
37 views

Question about the definition of a supremum

I want to know if the next definition is correct or I should fix it: Lets say that we have group $K$ and $a\in K$. Then, if $a\sup K$, for every $x\in K\Rightarrow x<a$? Or I shold cahge it to ...
0
votes
2answers
99 views

Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$

Hi guys this was a practice problem I was given, can anyone help me out on it? This is the problem: Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$ and the following is what I ...
1
vote
1answer
52 views

About a notation in Rudin's “Real and Complex Analysis”

On page 14. Suppose $\{f_n\}$ is a sequence of extended-real functions on a set $X$. Then $\sup\limits_nf_n$ and $\limsup\limits_n f_n$ are the functions defined on $X$ by $$ ...
0
votes
0answers
45 views

Demonstration of the Borel-Cantelli lemma

I am trying to understand the demonstration of the Borel-Cantelli lemma: Let $(A_n)$ be a sequence of independent events. If $ \sum_n P(A_n) = \infty $, then $P( \lim \sup A_n ) = 1$ To show ...
2
votes
2answers
247 views

Does $\lim \sup x_{n+1}-x_n=+\infty \implies \lim \dfrac{n}{x_n}=0$

If $(x_n)$ is a real sequence such that $\lim \sup x_{n+1}-x_n=+\infty$ , then must we have $\lim \dfrac{n}{x_n}=0$ ?
1
vote
1answer
47 views

Limit, Limsup clarification needed.

I know that if $(a_n)$ and $(b_n)$ are convergent sequences, and $a_n \leq b_n$, $\forall n\ge N$, then $\lim\, a_n\leq \lim\,b_n$. Now my question is what if we don't know that $(a_n)$ and $(b_n)$ ...
0
votes
1answer
41 views

$P(\liminf S_n)=1$ what does it mean?

$P(\cdot)$ is the probability measure. $S_n$ is a sequence of events. $P(\liminf S_n)=1$ does it mean that $S_n$ always happen after some large n? Can I say that it must be true that $\liminf ...
3
votes
0answers
24 views

Order of an entire $f $ is $\limsup_{r \rightarrow + \infty} \frac{\log \log M(r)}{\log r}$ [duplicate]

An entire function is of finite order $\rho$ if $$\rho = \inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ Write $M(r) = ...
2
votes
1answer
44 views

Limits of a map on the space of bounded sequences

I'm working on an exercise consisting of several questions that I can't quite figure out. If someone could give me any tips I'd be very happy! For $\xi = (x_1, x_2,...) \in \ell^\infty$ define ...
0
votes
1answer
29 views

Proof of Convergence in Distribution and Limsup

I'm currently using 'Adventure in Stochastic Processes' for self-study. Here's the link. This is the part I don't understand: Letting $\left[n\rightarrow \infty\right]$, we get $\limsup_{n\rightarrow ...
0
votes
1answer
79 views

Limes superior of product of two sequences

I'm an undergraduate mathematics student and I'm trying to understand the basics of limes superior. I recently got stuck on the following question: Let $(a_n)$ be a sequence of numbers and let ...
0
votes
2answers
103 views

Lim sup sequence

Let $(a_n)$ be a sequence of numbers. Show: If $(a_n)$ converges, than: $\lim\limits \sup a_n= \lim\limits_{n \rightarrow \infty} a_n $ I can feel this is true intuitively, but I have no idea how to ...
0
votes
3answers
72 views

What is a supremum?

I'm reading here about sequence of functions in Calculus II book, and there's a theorem that says: A sequence of functions $\{f_n(x)\}_0^\infty$ converges uniformly to $f(x)$ in domain $D$ ...
0
votes
1answer
100 views

lim sup of the square equals square of the lim sup

"Suppose that for the sequence of real numbers {an}, lim sup (an) = c > 0 Prove that lim sup (an^2) = c^2" For this question, I tried two ways: 1) Since c is the limsup of {an}, given e >0 and k>0, ...
1
vote
1answer
43 views

A limit involving the exponent of convergence

Let $f$ be an entire non-constant function with at least one zero. If $\{z_{j}\}_{j\in \mathbb{N}}$ are the zeros of $f$, set $$b =\inf\left\{\lambda >0 \ | \sum_{j}\frac{1}{|z_{j}|^{\lambda}}< ...
2
votes
1answer
72 views

Corollary of Banach Steinhaus theorem

If $\{M_n\}_{nāˆˆ\mathbb{N}}$ is a family of continuous operators for $X$ Banach to $Y$ normed, such that $M_n(x)$ converges to $M(x)$ for all $x āˆˆ X$, then $M$ is a linear bounded operator and ...
1
vote
1answer
31 views

$X_{n}$ and $Z$ be random variables if $X_{n} \ge Z$ then $ E[\liminf_{n\to \infty} X_{n}] \le \liminf_{n\to \infty} E[X_{n}] $

Let $X_{n}$ and $Z$ be random variables on probability space $(\Omega ,\mathcal F,P)$ and $Z$ be integrable. Show that $$X_{n} \ge Z \qquad \Longrightarrow \qquad E[\liminf_{n\to \infty} X_{n}] \le ...
5
votes
2answers
68 views

Proving a lower bound on the limit superior of a sequence.

Prove that for every positive sequence {$a_{n}$}, $$\varlimsup_{n \to \infty}\frac{\sum_{i=1}^{n+1}a_{i}}{a_{n}}\geq 4$$ Also find the sequences {$a_{n}$} for which 4 is attained. Attempted ...
3
votes
1answer
44 views

The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
1
vote
0answers
41 views

Can I conclude the following about limsup

I am trying to show that if $F:[a,b]\rightarrow\mathbb{R}$ is continuous and of bounded variation then $g(x)=\limsup_{h\rightarrow 0, h>0} \frac{F(x+h)-F(x)}{h}$ is a Lebesgue measurable function. ...
2
votes
1answer
89 views

Prove $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$

I'm having problems with the following proof: If $s_n$ and $t_n$ are sequences, then does $\liminf (s_n\cdot t_n) = \liminf (s_n) \cdot \liminf (t_n)$? Is there a theorem that proves this? Is this ...
1
vote
2answers
26 views

does taking limsup preserve inclusion relation?

Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of sets with no further structure at that point, such that $a_n \subset b_n$ for every $n\in \mathbb{N}$, does it holds that ...