# Limsup of independent exponential random variables

In the lecture, we are given an example on independent and identically distributed random variables, but I am not quite sure what is the idea of this question. The exercise is as following:

Let $$(X_n)$$ be a sequence of independent and identically distributed random variables defined on a probability space $$(\Omega,\mathcal{F},\mathbb{P})$$. Assume that $$X_1$$ is exponentially distributed with parameter $$\lambda > 0$$. The distribution of $$X_1$$ is absolutely continuous with density function $$f(x)=\lambda e^{-\lambda x}1_{(0,\infty)}(x)$$, $$x\in\mathbb{R}$$.

Show that $$\mathbb{P}(\lim\sup_{n\to \infty}X_n/\log(n)=1/\lambda)=1$$.

I know that for the exponential distribution, the mean $$\mathbb{E}[x]=1/\lambda$$, it says $$X_1$$ is exponentially distributed, is that mean all of the $$X_n$$ are also exponentially distributed? $$\lim\sup_{n\to\infty}(X_n)$$ means that $$X_n$$ happens for infinitely often times as $$n$$ close to infinity. so $$\lim\sup_{n\to\infty}(X_n)=\bigcap_{n∈N}\bigcup_{m>n}X_n$$, and also as $$n\to\infty$$, $$\log(n)\to\infty$$ . I am not sure about what this question is asking about? Is it something with $$0/1$$ law? like if an event $$A$$ happens infinitely times, then $$\mathbb{P}(A)\in\{0,1\}$$

– Alex
May 4, 2014 at 6:49
• I am not sure how to use Latex in here, sorry May 4, 2014 at 7:00
• @xz330 I have submitted a revision to your question that fixes the latex formatting. Please check that I haven't altered the meaning of your question. May 4, 2014 at 7:49
• Changed the title to something more accurate. Do you understand why it fits better the content of the question?
– Did
May 8, 2014 at 9:09

Show that $P(lim(n→∞)supXn/logn=1/λ)=1$.

(...) $lim(n→∞)(supXn)$ means that Xn happens for infinitely often times as n close to infinity. so $lim(n→∞)(supXn)=⋂_n⋃_mXn$, where m>n and n∈N.

This is confusing limsup of events and limsup of random variables. Here, each $X_n$ is a random variable hence $$\limsup\limits_{n\to\infty}X_n/\log n$$ is the random variable $Y$ such that, for every $\omega$ in $\Omega$, $$Y(\omega)=\limsup\limits_{n\to\infty}X_n(\omega)/\log n.$$ And the task is to prove that the event $[Y=1/\lambda]$ has probability $1$.

Note that the question most probably does not reproduce faithfully the text of the exercise asked. For example, $$\text{lim(n→∞)supXn/logn}$$ should read $$\limsup\limits_{n\to\infty}X_n/\log n,$$ where $\limsup\limits_{n\to\infty}$ acts as a single operation, or, equivalently, $$\limsup\limits_{n\to\infty}\frac{X_n}{\log n}.$$

• oh right, I got what it means now, but how do you prove that $P[Y=1/λ]=1$? May 4, 2014 at 7:58
• Yeah, I have changed all the equations to the proper formatting now. May 4, 2014 at 8:00
• That is a different question.
– Did
May 4, 2014 at 8:01
• isn't it what the question is asking about? May 4, 2014 at 8:01
• "If you want an answer to that other question, you should add your thoughts on it" (bis).
– Did
May 8, 2014 at 9:07