sum of Paretos
A rather curious question on X validated about the evolution of
![\mathbb E^{U,V}\left[\sum_{i=1}^M U_i\Big/\sum_{i=1}^M U_i/V_i \right]\quad U_i,V_i\sim\mathcal U(0,1)](https://s0.wp.com/latex.php?latex=%5Cmathbb+E%5E%7BU%2CV%7D%5Cleft%5B%5Csum_%7Bi%3D1%7D%5EM+U_i%5CBig%2F%5Csum_%7Bi%3D1%7D%5EM+U_i%2FV_i+%5Cright%5D%5Cquad+U_i%2CV_i%5Csim%5Cmathcal+U%280%2C1%29&bg=000000&fg=B0B0B0&s=0&c=20201002)
when M increases. Actually, this expectation is asymptotically equivalent to
![\mathbb E^{V}\left[M\big/\sum_{i=1}^M 2U_i/V_i \right]\quad U_i,V_i\sim\mathcal U(0,1)](https://s0.wp.com/latex.php?latex=%5Cmathbb+E%5E%7BV%7D%5Cleft%5BM%5Cbig%2F%5Csum_%7Bi%3D1%7D%5EM+2U_i%2FV_i+%5Cright%5D%5Cquad+U_i%2CV_i%5Csim%5Cmathcal+U%280%2C1%29&bg=000000&fg=B0B0B0&s=0&c=20201002)
or again
![\mathbb E^{V}\left[1\big/(1+2\overline R_{M/2})\right]](https://s0.wp.com/latex.php?latex=%5Cmathbb+E%5E%7BV%7D%5Cleft%5B1%5Cbig%2F%281%2B2%5Coverline+R_%7BM%2F2%7D%29%5Cright%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
where the average is made of Pareto (1,1), since one can invoke Slutsky many times. (And the above comparison of the integrated rv’s does not show a major difference.) Comparing several Monte Carlo sequences shows a lot of variability, though, which is not surprising given the lack of expectation of the Pareto (1,1) distribution. But over the time I spent on that puzzle last week end, I could not figure the limiting value, despite uncovering the asymptotic behaviour of the average.
Xi'an's Og 
May 12, 2022 at 5:32 pm
Et en appliquant
à 
on devrait y voir plus clair?
May 13, 2022 at 11:56 am
Pas vraiment…
May 15, 2022 at 10:59 am
Si
alors
et
Consequence
May 15, 2022 at 12:53 pm
Correction A(t)=(1-e^{-t})/t evidemment. Si on remplace les lois uniformes de $U_i$ et $V_i$ par deux lois sur la demi doite quelconques la valeur doit etre encore jolie
May 15, 2022 at 2:21 pm
Les simulations ne semblent pas concorder avec cette valeur limite, puisqu’elles indiquent plutôt 1/8 ?!
May 12, 2022 at 8:10 am
[…] article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here) […]