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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.

The response errors have standard deviations proportional to $|{\mu }_i|$. Thus, when fitted means approximate the true means, increasing absolute fitted values tends to accompany increasing residual spread. A residual-versus-fitted plot shows a fan for positive means, or spread increasing away from zero for means of either sign. Standardization by one common residual standard deviation, even with the usual regression leverage correction, does not remove this systematic heteroscedasticity.

Under independent errors the exact covariance matrix after fitting iswhere $k>0$ is the common proportionality constant. Hence the simple fan description is a diagnostic tendency, not an assertion that each fitted residual has exactly variance $k{\mu }_i^2$.

For positive responses, the intended variance-stabilizing transformation is the natural logarithm. The delta method givesUse $\log Y_i$ for positive data with modest relative error, then refit and check the transformed mean and residual structure. In particular, if the original mean was $X_i^T\beta$, the transformed mean is approximately $\log (X_i^T\beta )-k/2$, not automatically a linear function of the original covariates.

There is a domain qualification in the literal printed model: a nondegenerate normal distribution has negative support. For positive ${\mu }_i$, the probability of $Y_i\le 0$ is $\Phi (-1/\sqrt{k})$, so a real $\log Y_i$ is not defined throughout the stated sampling model. When $k$ is small this is a negligible-tail approximation, not an exact transformation to normal errors. If signed values must be retained and all ${\mu }_i\ne 0$, write $Y_i={\mu }_i(1+\sqrt{k}{Z}_i)$ with standard normal $Z_i$. ThenThe second term has a common distribution with finite variance, so log absolute value stabilization of a normal scale family is exact, but loses the sign of the mean and does not produce normal errors. Zeros and a zero mean are excluded from this transform. Alternatively retain the response scale and fit the stated variance model with weighted least squares weights proportional to $1/{\hat{\mu }}_i^2$ where the means stay away from zero. No arbitrary addition of a constant before taking logarithms exactly solves the given variance relation.