= Additivity of independently driven squared Bessel processes
{title2=$\operatorname{BESQ}^{\alpha}+\operatorname{BESQ}^{\beta}\sim\operatorname{BESQ}^{\alpha+\beta}$}
For independently driven nonnegative <squared Bessel processes> $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is <Brownian motion> by the <Lévy characterization of Brownian motion>. Multiplying by $2\sqrt Z$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha+\beta$.
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