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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.

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  • $\begingroup$ 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
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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?

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  • $\begingroup$ 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

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