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If we let $\underset{k\times 1}{\boldsymbol{X}}=(X_1, \dots, X_k)' \sim MP^{(k)}(\boldsymbol{0},\boldsymbol{1}, \alpha)$ where MP denotes a Multivariate-Pareto distribution, with joint survival function:

$F_{\boldsymbol{X}}(x_1, \dots x_k)=\left(1+\sum_{i=1}^k (x_i-\mu_i)/\sigma_i\right)^{-\alpha}=\left(1+\sum_{i=1}^k x_i\right)^{-\alpha}$,

where the normalizations $\underset{k\times 1}{\boldsymbol{\mu}}=(\mu_i)=(0,\dots, 0)'=\boldsymbol{0}$ and $\underset{k\times 1}{\boldsymbol{\sigma}}=(\sigma_i)=(1,\dots, 1)'=\boldsymbol{1}$, have been imposed above.

Then, $Y=\sum_{i=1}^k X_i \sim FP(0,1,1,\alpha, k)$ where FP denotes a Feller-Pareto random variable.

I have seen this result in numerous places without proof: e.g., H.C. Yeh, "Some Properties of Homogeneous Multivariate Pareto Distribution", Journal of Multivariate Analysis, (1994) and B. Arnold's text, "Pareto Distributions 2nd Edition" Ch.6; probably because it is straight forward. Both references simply mention that by Jacobians, the result follows.

But I am having a hard time because if $y=g(X_1,\dots X_k)$ I am not sure how to obtain $g^{-1}(y)$. I believe some auxiliary variables need to be introduced? Any ideas? I am also wondering what happens if we let $k\rightarrow \infty$. By inspection of the density of the FP, I believe the result breaks down.

yungmist
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  • Because $g^{-1}$ makes no sense when $k\gt 1,$ could you explain what you mean by that? – whuber May 18 '22 at 18:12
  • @whuber Yes, I agree. Thats exactly where my confusion lies. If $k=1$, then I don't have any confusion about $g^{-1}$, although that would change the problem/result. I brought up $g^{-1}$ because both references mentioned "proof via Jacobians", which I assumed requires $g^{-1}$. – yungmist May 18 '22 at 18:20
  • Jacobians arise naturally in multivariate integrals and do not require inverses. See https://stats.stackexchange.com/a/154298/919 for one account of this. But your situation looks particularly simple, since $Y$ is univariate: all you need to is compute $\Pr(Y\le y)$ for positive real numbers $y.$ – whuber May 18 '22 at 18:56
  • For computing $P(Y \leq y)$, am I supposed to know the density $f_{X_1+\dots + X_k}(y)$ or am I supposed to apply a change of variables? Since the $X_i$'s are not i.i.d. I don't think I know how to characterize that density directly. – yungmist May 18 '22 at 21:22
  • Learned a lot from your referenced post, that technique is quite remarkable. Thank you for the reference. As you said, this problem appears easier than that since $Y$ is univariate. On the other hand, not sure how to adapt that technique to this problem since $X_i$'s are not i.i.d, which is where your preliminaries section starts the discussion in the referenced post. – yungmist May 19 '22 at 00:16
  • @whuber Any follow up help? – yungmist Jun 01 '22 at 17:28
  • Sorry, I missed this. Nope. I am still quite curious to close out this problem though. – yungmist Aug 26 '23 at 05:39

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