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Let $\{X_n\}$ be a sequence of independent Exponential random variables with mean $$ E(X_n)=\frac{1}{\lambda_n}, $$ where $$ 0 < \lambda_n < \infty. $$ If $$ \sum \frac{1}{\lambda_n} = \infty, $$ show $$ P\left(\sum X_n = \infty\right) = 1. $$

It looks like a job for the Borel-Cantelli lemma, but I don't see which sequence of events will give the result.

Let $$ X=\sum X_n, $$ then $$ \sum \frac{1}{\lambda_n}=E(X) \ge \sum P(X>n), $$ but this is in the wrong direction.

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up vote 1 down vote accepted
  1. The event $\sum_j X_j=\infty$ is a tail event, hence its probability is $0$ or $1$.
  2. Assume it is $0$; then the series $\sum_j X_j$ is almost surely convergent, hence it is supposed to converge in law.

  3. Computing the characteristic function of partial sums, and in particular their modulus, one can see this can't happen.

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Whoa! Of very great level. Thanks. – Nicolas Essis-Breton Apr 29 '13 at 20:25
You are welcome. – Davide Giraudo Apr 29 '13 at 20:30

I can't reach the contradiction. I get the characteristic function of the partial sums to be $\phi_{n}(\theta) = \prod_{i=1}^{n} \left(\frac{1}{1-i\frac{\theta}{\lambda_{i}}} \right)$. So $|\phi_{n}(\theta)| = \prod_{i=1}^{n} \frac{1}{\sqrt{1 + (\frac{\theta}{\lambda_{i}})^2}} \leq 1$. What am I doing wrong?

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