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Derive the 0-1 law from a functional equation of a martingale limit

For $z>0$ and $0<x<1$ you have $$ \lim_{t\to+\infty}\Bbb E(e^{-tM(z)})=\lim_{t\to+\infty}\Bbb E(e^{-tzx^{2z-1}M(z)})=\lim_{t\to+\infty}\Bbb E(e^{-tz(1-x)^{2z-1}M(z)})=\Bbb P(M(z)=0). $$ ...
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