If I take the sample mean of a scaled $\chi$ distribution of $N$ samples, the distributions of these means should lead to a normal distribution $\mathcal{N}(\mu,\sigma)$. This procedure is repeated $M$ times.
Now I want to normalize the data sample wise ($M$) by the mean of each of the $M$ samples. The mean of the distribution of means should also follow a normal distribution $\mathcal{N}(\mu,\sigma/\sqrt{M})$. I order to charaterize the resulting ratio (determine thresholds) I'd like to used a transformation to a standard normal distribution (Hinkley, Biometrica 1969, 'On the Ratio of Two Normal Random Variables'). This approach requires knowledge about the correlation factor between both distributions.
Simulations showed me, that the correlation between the sample means ($N \times M$) and corresponding means over $N$ is $1/\sqrt{M}$. How can I prove that?