Distribution of the sum of $m$ inverse exponential random variables

Question

Let $T_1, \ldots, T_m$ be $m$ random variables with $T_i \sim \exp(\lambda_T)$. We are interested in the sum

$$ T' = \sum_{i=1}^{m}\frac{1}{T_i} $$

which is itself a random variable. By definition, each term in the sum follows an inverse exponential distribution with parameter $\lambda_T$. However, I have found no literature covering the distribution of a sum such as this one. I have two questions:

$a.$ Can one derive the distribution of $T'$ analytically? If so, how?

$b.$ Is there any literature covering the sum of inverse exponential random variables?

Thanks in advance.

Answer 1

The inverse exponential variable $Y_i = 1/T_i$ has a Fréchet distribution with PDF $$f_Y(y) = \frac{\lambda e^{-\lambda/y}}{y^2}, \quad y > 0.$$ However, the sum of iid $Y_i$ does not have an elementary closed form, since even the case $n = 2$ requires the evaluation of $$f_{T'} (t) = \int_{y=0}^t f_{Y_1}(t - y) f_{Y_2}(y) \, dy = \lambda^2 \int_{y=0}^t \frac{e^{-\lambda(1/(t-y) + 1/y)}}{(y(t-y))^2} \, dy.$$ Needless to say, even the case $\lambda = 1$ is intractable.

Moreover, the mean is infinite. I suspect the absence of literature on the topic may be due to these properties--no closed form and no first moment or higher. Moreover, what kind of random process is modeled by such a statistic?