 Log absolute value stabilization of a normal scale family

For $Y\sim N(\mu ,k{\mu }^2)$ with $\mu \ne 0$ 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 ${\int }_0^1|\log t{|}^{2}dt<\infty$, 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.