I am curious as to what describes the following distribution. If we were to record some data which are all from a normal distribution, but the standard-deviation changes for blocks of points recorded. For example, in the following plots we can see normally distributed data where each colour represents a different values of $\sigma$.
In the histogram all of the data have been plotted, which forms something that vaguely resembles a Student-T distribution -- although I don't think this is representative. I have also plotted the PDF's of each parent distribution on top of the histogram.
It clearly isn't a convolution of Normal distributions as the convolution of normal distributions, should itself be a normal distribution.
If I however sum the four normal distributions, the result perfectly envelopes the entire histogram:
That is to say $$\frac{1}{N}\sum_{i = 1}^{N}G(\mu, \sigma_{i})$$ where each $\sigma_{i}$ represents different standard deviation. In the example I have shown here $\mu = 0$ for all cases.
Is there a name for this distribution?