= Gaussian characterization by independent sum and difference
{c}
{title2=$X+Y\perp X-Y$}
Here take $X,Y$ iid, centered with <variance> one, and with finite <moment-generating functions> at all real arguments. Independence of their sum and difference gives $M(2t)=M(t)^3M(-t)$. Dividing the positive and negative argument identities shows $M(t)/M(-t)$ satisfies the symmetry step of <dyadic rigidity of a moment-generating function>, hence $M$ is even. The identity becomes $M(2t)=M(t)^4$, and the unit-variance expansion then forces $M(t)=e^{t^2/2}$. The <uniqueness theorem for moment-generating functions> identifies the <standard normal distribution>. This is a precise sufficient-hypothesis version; the displayed implication is understood with those stated assumptions.
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