Oh FFS read the thread.
In a self-similar system, the two axes scale together at some fixed rate. This does not mean the amplitudes are the same at all scales, but that they vary consistently. Capeldodger argued it was a necessary condition of self-similarity that one axis would be fixed as the other scaled, i.e. that you would see the same amplitude of variation day-to-day as you would millenium-to-millenium. This is fundamentally wrong.
And I didn't say it. What I said was that the shape of the graph remains the same at any scale, in a self-similar system. This is clearly not the case for the hydrological cycle (which includes ice).
Another feature is that any part of the system looks like any other part, which again is not the case. A millennium during a glaciation looks different from a millenium during an inter-glacial, and both look different from a millennium during a transition. Even the transitions to an inter-glacial and from an inter-glacial look very different - the former is rapid and pretty wild, the latter is gradual and reserved. That's because it's easier for the system to lose ice suddenly than to accumulate it suddenly.
Your "stochastic" model does not sit well with the observed regularity of ice-ages. You've previously claimed that these can't be explained by Milankovich cycles and feedback models without them becoming wildly erratic, but that is simply not so. The feedbacks involved - albedo and greenhouse gases - are physically limited. You can't go less than no ice-cover, and there's only so much CO2 in the system, so the models cannot run out of control.
The albedo feedback in itself is not sufficient, although it was a fine insight. Add in the CO2 exchange between ocean and atmosphere (and the water-vapour feedback in the atmosphere) and Milankovich is justified. So much for ice-ages - the major field of climate science until very recently
What's new on the short time-scale is that lots of new CO2 is being stuffed into the system. (By "short" I mean at least this ice-epoch, four million years or so.) Far from there being an exchange of CO2 between ocean and atmosphere there's a continuing injection of CO2 into both. A whole new situation, and statistical comparisons with other situations ain't worth squat.
Working out the ice-ages was a triumph in itself, but the possible implications for this new situation weren't missed. Arrhenius had mused that AGW was plausible centuries down the line (that was before the automobile age), Angstrom's lab had pissed on that bonfire in the 20's, their mistake was discovered in the 40's, and the ice-age problem was solved. Which did bring Arrhenius's thought back into focus - what would be the impact of a substantial artificial introduction of CO2 into the atmosphere?
What started as a niggling question fifty years ago has turned into a reality. Try as you might to believe that this is all normal.