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I am trying to understand the behavior of the following integral

$$\int_{0}^{\infty}\prod_{i=1}^{N/2}\left( e^{-\lambda_{2i-1}t}+e^{-\lambda_{2i}t}-e^{-(\lambda_{2i-1}+\lambda_{2i})t} \right)dt$$

given $\lambda_1,\dots,\lambda_N$. What I'm specifically interested in is the decay of this integral with increasing $N$ (i.e. when more $\lambda$'s are added). I tried obtaining a recurrence relation using integration by parts but couldn't get anywhere. I don't need an exact expression but just need to determine how adding more $\lambda_i$'s impacts the integral, so any ideas on approximating this integral or finding an approximate recurrence pattern are also welcome.

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  • $\begingroup$ Is there any assumption on $\lambda_i's$? One thing that I can prove is that the integral remains bounded away from $0$ if $\sum_{i=1}^{\infty} \lambda_{2i-1}\lambda_{2i} < \infty$. Also, given that the integran can be written as $$\mathbf{E}\left[\min_{1\leq i \leq N/2} \max\{X_{2i-1}, X_{2i}\} \right]$$ for independent RVs $X_i \sim \mathrm{Exp}(\lambda_i)$, some probabilistic approach may be available, although I haven't thought about it yet. $\endgroup$ – Sangchul Lee Apr 19 at 17:28
  • $\begingroup$ The $\lambda_i$'s individually are small, but they may not necessarily be decreasing, so the sum of $\lambda_{2i-1}\lambda_{2i}$ is probably not bounded. The expected value you wrote is exactly how I got to this integral, so any ideas using the probabilistic approach are also welcome. What I really want to figure out (an approximation would be fine as well) is how the expected value changes from $N=k$ to $N=k+1$ (i.e. the effect of adding another pair $X_{2k+1}$ and $X_{2k+2}$). $\endgroup$ – Zaeem Hussain Apr 19 at 18:26

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