So assume there's a normal distribution of price possibilities at each purchase point. You're going to end up with the _sum_ of those distributions, not the correlation or combination of them.

Presumably the deviations of those are growing more larger over time, but the are still the same underlying distribution curve as they sum up. The mean will trend steadily upwards.

The resulting price will be a normal distribution with a deviation of somewhere around the midpoint, and a mean of somewhere around the midpoint.

Now, with the mean increasing over time, your expected return will be decreasing the longer you wait. But what about the risk:return?

Unless you have outside knowledge about risk growing or lessening over time, it will remain constant over the DCA period, which means the risk factor of your investment (the standard deviation of the resulting DCA price) will be at exactly the midpoint of your purchase series.

So your expected return will be slightly lower, and your risk profile will be substantially larger, than just buying immediately.

DCA: math does not check out, not even close, _if you have all the money upfront_.