I'm studying for an exam at the moment, and this type of questions has just got me stumped to the point where I need a step-by-step walkthrough.

More specifically, I've got one problem I just can't get past:

Let $(X_n)$ be i.d.d random variables. $X_n$ ~ $Exp(1)$. Denote $Z_n = min\left\{X_1, X_2,..., X_n\right\}$. Prove that $Z_n$ converges almost surely to $0$.

I'm guessing the first step is to find $Z_n$'s distribution function, but I don't know what to do next. Please help. Thanks in advance.

  • $\begingroup$ The first thing to prove is that the minimum of $n$ independent identical exponentially distributed random variables with parameter $1$ is itself exponentially distributed with parameter $n$. In fact, the parameters can even be different, say $\lambda_1,\dotsc,\lambda_n$; as long we keep independence then the minimum is distributed like $\exp(\sum \lambda_i)$. Knowing this, it should be easier to show $Z_n \to 0$ almost surely. $\endgroup$ – Nap D. Lover Apr 17 '17 at 14:01
  • $\begingroup$ Actually, my problem is how to show $Z_n → 0$ almost surely. :( $\endgroup$ – Sophie Apr 17 '17 at 14:11
  • $\begingroup$ I understand. If you know the distribution of $Z_n$ then you should be able to finish Lord Shark's answer below. $\endgroup$ – Nap D. Lover Apr 17 '17 at 14:13

The sequence $Z_n$ is nonnegative and decreasing: $Z_1\ge Z_2\ge Z_3\ge \cdots\ge0$. For it to fail to converge to zero, there will be $N\in \Bbb N$ with $Z_k\ge 1/N$ for all $k$ (equivalently, $X_k\ge 1/N$ for all $k$). What's the probability of that?

  • $\begingroup$ Can you write it more specifically? I understand your point but I don't know how to express it clearly! Thanks! $\endgroup$ – Sophie Apr 17 '17 at 16:43
  • $\begingroup$ @Sophie what exactly are you having trouble expressing clearly? What you need to do is calculate $P(Z_n \geq 1/N)$ and show this $\to 0$ as $N \to \infty$. If calculating the probability before taking the limit is giving you trouble then I infer you are confused about the distribution of $Z_n$, which my first comment addresses and you should reread carefully. If not, then you need to specify what is impeding your work. $\endgroup$ – Nap D. Lover Apr 17 '17 at 17:22
  • $\begingroup$ @Sophie I've amplified my solution a bit: the point is to compute the probability of the event $\{X_k\ge 1/N \textrm{ for all }k\in\mathbb{N}\}$. $\endgroup$ – Angina Seng Apr 17 '17 at 17:26
  • $\begingroup$ I got it! Thanks. $\endgroup$ – Sophie Apr 18 '17 at 11:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.