the stability index alpha
Among the four parameters of a stable law (index, skewness, scale, location), one governs everything qualitative: the stability index alpha. It is the single number that fixes the tail heaviness, which moments exist, the correct normalization for sums, and whether the LLN and CLT survive. Knowing alpha is knowing the regime.
The index alpha ranges over (0, 2]. It enters the stable characteristic function as the exponent in -|sigma t|^alpha, and equivalently it is the tail exponent: for alpha < 2, P(|X| > x) ~ c x^(-alpha) as x -> infinity, a power law (regularly varying with index -alpha). The consequences are mechanical. Moments: E[|X|^p] < infinity exactly when p < alpha (with alpha=2 the boundary normal case having all moments). So alpha > 1 gives a finite mean, alpha = 2 a finite variance, alpha <= 1 not even a finite mean. Sum scaling: for iid stable summands, S_n = X_1 + ... + X_n satisfies S_n / n^(1/alpha) being distributed like X (up to centering), so the spread grows like n^(1/alpha) — faster than the Gaussian sqrt(n) = n^(1/2) whenever alpha < 2.
Alpha matters because it diagnoses which limit theory applies. For alpha = 2 you are in the classical Gaussian world (finite variance, sqrt(n) scaling, CLT). For 1 < alpha < 2 the mean exists and the LLN holds, but variance is infinite and sums converge to a non-Gaussian stable law under n^(1/alpha) scaling. For alpha <= 1 even the LLN fails. The honest pitfall: people fit a Gaussian to heavy-tailed data and dramatically underestimate extreme risk because they implicitly assume alpha = 2; estimating alpha from tail data (Hill estimator) is delicate and the n^(1/alpha) scaling is genuinely different — for alpha = 1.5, sums spread like n^(2/3), much wider than the Gaussian n^(1/2).
If X has alpha = 1.5, then E[X] exists (since 1 < alpha) but E[X^2] = infinity (since 2 > alpha). Sums of n iid such variables, recentred, spread out like n^(1/1.5) = n^(2/3) rather than n^(1/2) — so a Gaussian fit would underestimate how much large sums fluctuate.
alpha sets the tail exponent, which moments exist (p<alpha), and the n^(1/alpha) sum scaling.
alpha=2 is the only stable case with finite variance; for alpha<2 fitting a Gaussian severely underestimates tail risk and the correct sum normalization is n^(1/alpha), not sqrt(n).