I am exploring whether a linear combination of power law distributions is also a power law distribution. Specifically, if $X \sim (\alpha -1) x_{\min}^{(\alpha -1)} x^{-\alpha}$ and $Y \sim (\beta -1) y_{\min}^{(\beta -1)}y^{-\beta}$, we are interested in finding the distribution of $Z = \rho X + (1-\rho) Y, \rho \in (0, 1)$? ($X$ and $Y$ are independent)
I tried with the moment generating function approach:
If $M_Z(t)$ is the moment generating function of $Z$, $$M_Z(t) = E[e^{tZ}]=E[e^{t(\rho X + (1-\rho)Y}]=E[e^{t\rho X}]\cdot E[e^{t(1-\rho)Y}] = M_X(t\rho) M_Y(t(1-\rho)).$$
This brings us to the moment generating functions of the power law distributions themselves:
$$M_X(s) = \int_{x_{\min}}^\infty e^{sx}(\alpha -1)x_{\min}^{(\alpha-1)} x^{-\alpha} \, dx = (\alpha – 1)x_{\min}^{(\alpha-1)}\int_{x_{\min}}^\infty e^{sx}x^{-\alpha} \, dx. $$
Like it is mentioned here, this resembles the incomplete gamma distribution, but I cannot complete the derivation. And, more importantly, I am not sure if this will help me answer the original question, i.e., is the linear combination of two power law distributions also a power distribution?
Best Answer
The moment generating function doesn't exist for many heavy tailed distributions. In general if you take suitably normalized linear combinations of power-law distributed random variables with infinite variances (ie with tail parameter $\alpha<2$, the generalized version of central limit theorem shows weak convergence to a stable distribution with the same tail parameter. If $\alpha\geq 2$ the linear combinations will tend to converge toward a Gaussian. Note that when $\alpha=2$ the variance may still be infinite but convergence to Gaussian is still possible. An example of a heavy tailed stable distribution is the Cauchy. All stable distributions except Gaussian have power law tails. John P. Nolan has some useful notes on his website.