Stable size-biasing and the positive scale-mixture order of generalized Gaussian laws

Abstract

Let $X_r\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\stackrel{d}{=}VX_q$ if and only if $p\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $α=b/a=p/q$. If $S_α$ is a positive $α$-stable random variable with Laplace transform $\mathbb{E}\exp(-uS_α)=\exp(-u^α)$, set $W_0=S_α^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and define $V_{p,q}=2^{a-b}W$. Then $X_p\stackrel{d}{=}V_{p,q}X_q$. For $p>q$, the required Mellin quotient, viewed as the candidate characteristic function of $\log V$, is unbounded by Stirling&#39;s formula, and hence cannot be a characteristic function. The factor laws form a multiplicative cocycle, $V_{p,r}\stackrel{d}{=}V_{p,q}V_{q,r}$, for $p\le q\le r$, where the factors on the right-hand side are independent copies. Thus the Mellin quotient isolated by Dytso, Bustin, Poor and Shamai is realized constructively throughout the $p<q$ branch. In particular, $Φ_{p,q}$ is positive definite exactly in the range $p\le q$, and the inverse Fourier--Mellin candidate density in the remaining $p<q$ branch is a genuine nonnegative probability density. The known Gaussian-base and bounded-parameter product cases are recovered as parts of a single positive scale-mixture classification.