## Probability of the maximum (Levy Stable) random variable in a list being greater than the sum of the rest?

Given a list of identical and independently distributed Levy Stable random variables, $(X_0, X_1, \dots, X_{n-1})$, what is the is the probability that the maximum exceeds the sum of the rest? i.e.:

$$M = \text{Max}(X_0, X_1, \dots, X_{n-1})$$ $$\text{Pr}( M > \sum_{j=0}^{n-1} X_j - M )$$

Where, in Nolan's notation, $X_j \in S(\alpha, \beta=1, \gamma, \delta=0 ; 0)$, where $\alpha$ is the critical exponent, $\beta$ is the skew, $\gamma$ is the scale parameter and $\delta$ is the shift. For simplicity, I have taken the skew parameter, $\beta$, to be 1 (maximally skewed to the right) and $\delta=0$ so everything has its mode centered in an interval near 0.

From numerical simulations, it appears that for the region of $0 < \alpha < 1$, the probability converges to a constant, irregardless of $n$ or $\gamma$. For $1 < \alpha < 2$ it appears to go as $O(1/n^{\alpha - 1})$ (maybe?) irregardless of $n$ or $\gamma$. For $\alpha=1$ it's not clear (to me) but appears to be a decreasing function dependent on $n$ and $\gamma$.

I have tried making a heuristic argument to the in the form of:

$$\text{Pr}( M > \sum_{j=0}^{n-1} X_j - M) \le n \text{Pr}( X_0 - \sum_{j=1}^{n-1} X_j > 0 )$$

Then using formula's provided by Nolan (pg. 27) for the parameters of the implicit r.v. $U = X_0 - \sum_{j=1}^{n-1} X_j$ combined with the tail approximation:

$$\text{Pr}( X > x ) \sim \gamma^{\alpha} c_{\alpha} ( 1 + \beta ) x^{-\alpha}$$ $$c_{\alpha} = \sin( \pi \alpha / 2) \Gamma(\alpha) / \pi$$

but this leaves me nervous and a bit unsatisfied.

Just for comparison, if $X_j$ were taken to be uniform r.v.'s on the unit interval, this function would decrease exponentially quickly. I imagine similar results hold were the $X_j$'s Gaussian, though any clarification on that point would be appreciated.

Getting closed form solutions for this is probably out of the question, as there isn't even a closed form solution for the pdf of Levy-Stable random variables, but getting bounds on what the probability is would be helpful. I would appreciate any help with regards to how to analyze these types of questions in general such as general methods or references to other work in this area.

Note: There didn't seem to be much interest from this site so I've cross-posted to math.stackexchange.com here.

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When $\alpha>1$, the LLN implies that the sum is nearly $n\E X$, and $\E X$ depends on $\delta$. If $\E X<0$ the probability tends to 1. If $\E X>0$ the probability decays as you say.
When $\alpha<1$ the probability indeed depends to a constant. You can estimate $\P(X_0>X_1+\dots+X_{n-1})$, since the sum is just an independent stable variable. These events are almost disjoint. In the maximally skewed case, the variables are positive, and then the events are completely disjoint.