# 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.

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### Show a $\liminf$ statement involving probabilities

Take a sequence of binary random variables $(Y_t)_t$ such that $$(A) \quad \Pr\Big(\lim_{T\rightarrow +\infty} \frac{1}{T}\sum_{t=1}^T Y_t=\nu\Big)=1.$$ Consider another sequence of discrete random ...
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### Series with $\lim \sup$ and $\lim \inf$ being $+$ and $-\infty$

Do you have an example of a series whose partial sums' $\lim \sup = \infty$ and whose partial sums' $\lim \inf = -\infty$ ? I feel like it's like repeatingly adding a whole bunch of positive terms ...
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### How to prove that $L^- \le c \le L^+$. Where $c$ is a limit point and $L^-$ and $L^+$ are the limit inferior and limit superior respectively.

I have been trying to prove this exercise for way too long and am nowhere near a proof. I have asked this question before and haven't got a satisfactory answer. This could be my own fault for not ...
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### Prove that $\varliminf_{n \to \infty}\left(\inf_{x\in X} f_n(x)\right) \leq \inf_{x\in X}\left( \varliminf_{n \to \infty} f_n (x)\right)$

We also know that $f_n(x)$ is bounded for every $n\in\mathbb{N}, x\in X$. Firstly using liminf definition I rewrote $\varliminf_{n \to \infty} f_n(x) = \lim_{n \to \infty} \inf_{k\geq n} f_k(x)$ And ...
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### Inequality involving changing order of limits and probability

I read this paper, in Corollary 1 the author claims that $$\underset{\pi \in [0, 1]}{\sup}\ W_T(\pi) \overset{p}{\to} \infty$$ as $T \to \infty$. Where $W_T(\pi)$ is Wald statistics but I think it ...
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### A question on upper and lower derivatives of $F$ on $[a,b]$

This question is from the book The Integrals of Lebesgue, Denjoy, Perron, and Henstock. I'm currently reading part of the book The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Slight ...
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Let $f:(0,+\infty) \to \mathbb{R}$, a continous and non negative function and $g:(0,1) \to \mathbb{R}$ a non negativa continuous function, if for all sequence $(t_n)$ such that $t_n\to +\infty$ \... 1 vote 2 answers 71 views ### show that \lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty How do i show formally that : \lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty. I know that if n is odd, then lim inf will be -\infty and if n is even then lim sup will be \infty. However, i dont ... 0 votes 0 answers 42 views ### Exchanging limsup with liminf I am working on a problem where I think I might be able to complete my argument if I can show the following relation. \begin{align*} \limsup_{k \rightarrow \infty} \liminf_{x' \rightarrow x} \frac{1}{... 5 votes 1 answer 136 views ### Show that \lim\limits_{n\to\infty}D_n exists if and only if \lim\limits_{n\to\infty}E_n=\emptyset This is one of the exercises of Halmos's measure theory book. I know it has been already treated but I've been asked to compute it by going via indicator functions. Exercises goes as follows: Let (... 0 votes 1 answer 39 views ### Equivalent defintion of \liminf_{s \to t} f(s)? Let D \subset \mathbb{R} and f: D \to \mathbb{R} and t a limit point of D. Is the definitionL:=\liminf_{s \to t} f(s):= \lim_{\epsilon \to 0} (\inf \{ f(s)| s \in D \cap (t-\epsilon,t+\... 50 views

### Swap $\limsup$ and $\mathbb{E}$ of sequence of functions evaluated at a random variable?

I am a postgrad with more of a background in the functional analysis point of view of things but am recently needing to get the expectation involved and it has been a few years since I've done much ...
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### $\lim\inf x_n$ can possibly be $-\infty$ or $\infty$? Either or Both?

Show that $\lim\inf x_n$ always exists for any sequence $x_n$ of real numbers and can possibly be $-\infty$. Proof for the existence part is easy if I assume the sequence is bounded finitely. It ...
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### The liminf and limsup in terms of accumulation points

Let $(x_n)_{n\in\mathbb N}$ be a sequence of real numbers. If it is bounded, then the Bolzano–Weierstrass theorem tells us that the set of accumulation points in $\mathbb R$ is non-empty. Furthermore, ...
### Optimal bounds for the product of the divisor function $d(n)$ in short intervals
Let $d(n)$ denote the number of divisors of a positive integer $n$. It is pretty obvious that $d(n) \ge 2$ for any given number $n \ge 2$, since every number is divisible by $1$ and itself. $2$ is ...