= Log absolute value stabilization of a normal scale family
{title2=$\log|Y|=\log|\mu|+\log|1+\sqrt{k}Z|$}
For $Y\sim N(\mu,k\mu^2)$ with $\mu\ne0$ and fixed $k>0$, represent $Y=\mu(1+\sqrt{k}Z)$ with $Z$ having the <standard normal distribution>. Taking the <natural logarithm> of the absolute value gives the displayed identity. Its random second term has a common distribution, independent of $\mu$, so this <variance-stabilizing transformation> gives an exactly constant <variance>. That variance is finite: the normal density is bounded near zero and $\int_0^1|\log t|^2dt<\infty$, while normal tails control logarithmic growth at infinity. The transformation is defined almost surely, loses the sign of the mean, and does not produce an exact <normal distribution>. A zero mean makes the original observation identically zero and is outside its domain.
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