# Questions tagged [limsup-and-liminf]

For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.

1,462 questions
Filter by
Sorted by
Tagged with
21 views

35 views

34 views

### Doubt on limit of a sequence of random variables

Let's define a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and a sequence of random variables $(X_n)_{n\geq0}$ defined on it. If I assume that the limit of such a sequence of random ...
26 views

41 views

44 views

### Prove almost surely that a series is unbounded

Let $\{X_n\}_{n\ge1}$ be a sequence of independent and identically distributed random variables. Let $S_n = \sum_{n=1}^NX_n$ for $N\ge1$. Now assume that $X_n$ takes values $1, 0 \text{ or} -1$ with ...
57 views

### If $a_n\leq b_n$ then $\limsup a_n\leq \liminf b_n$

If $a_n\leq b_n$ then $\limsup a_n\leq \liminf b_n$ This is a statement in Royden, but I am not sure how to prove it.
58 views

44 views

### Is it trivial that $\limsup\limits_{n\to\infty}\sqrt[n]{|a_{n+1}|}=\limsup\limits_{n\to\infty}\sqrt[n]{|a_{n}|}$

Is it correct that $\underset{n\to\infty}{\limsup_{n\to\infty}\sqrt[n]{|a_{n+1}|}=\limsup}\sqrt[n]{|a_{n}|}$? i know its correct for regular $\lim$ but im not sure for $\limsup$. Also, is it ...
21 views

### Prove the existence of two constant to bound a function

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a continous function, such as exists $p\in\mathbb{R}^+$, $p>1$ verifying: $$\liminf_{x\to\infty}\frac{f(x)}{|x|^p}=L\in(0,+\infty]$$ Prove the existence ...
23 views

12 views

### Is the bound of different quotients the same as that of weak derivatives?

In $\S 5.8.2$ of Evan's PDE, there is a theorem relating to different quotients and weak derivatives. Theorem 3 (ii) Assume $1 < p < \infty$, $u \in L^p(V)$, and there exists a constant $C$ ...
57 views

### When is $\limsup_{n \to \infty} (a_n+b_n) = \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$?

I know that $$\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n +\limsup\limits_{n \to \infty} b_n.$$ But what should apply to A and B if we should have "=" ? I can't find ...