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

Let $X_r\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\exp (-\mathrm{\mid }x{\mathrm{\mid }}^r\mathrm{/}2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_{p}V{X}_q$ if and only if $p\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1\mathrm{/}p$, $b=1\mathrm{/}q$, and $\alpha =b\mathrm{/}a=p\mathrm{/}q$. If $S_{\alpha }$ is a positive $\alpha$-stable random variable with Laplace transform $\mathbb{E}\exp (-u{S}_{\alpha })=\exp (-u^{\alpha })$, set $W_0=S_{\alpha }^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and define $V_{p,q}=2^{a-b}W$. Then $X_p{V}_{p,q}{X}_q$. For $p>q$, the required Mellin quotient, viewed as the candidate characteristic function of $\log V$, is unbounded by Stirling’s formula, and hence cannot be a characteristic function. The factor laws form a multiplicative cocycle, $V_{p,r}{V}_{p,q}{V}_{q,r}$, 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, ${\mathrm{\Phi }}_{p,q}$ 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.