Normally we discuss winrates in terms of BB/100 or BB/hour and stddev/100 or stddev/hour, and then we basically assume a normal distribution to run variance simulators, and determine things like risk of ruin or to get an idea of maximum expected downswings.

I'm just wondering how appropriate is this approximation for a winning player, in general. That is, when talking about things like expected risk of ruin, expected downswings, etc, or stuff like confidence intervals? As a winning poker player, what kinds of differences might we expect given our real distribution vs the normal approximation?

FWIW, I do understand that the central limit theorem says a sum of RVs will *eventually* approximate a normal distribution - but not sure how close we are to that eventuality with the types of questions that might concern a poker player.

Do you have a lot of hands in pokertracker or something similar? If so, try something like packaging hands into bundles of 100 or 1000, summing your wins for each bundle. There are descriptions of ways to test for normality here
https://en.wikipedia.org/wiki/Normality_test
but frankly if you make a histogram of these results, it's going to "look" pretty normal

Nope, just live results (which probably tend to look a bit different shape-wise than online results, I'd think)... This was more just a question out of curiosity. Like is it likely for a good player to lose more than a simulation (based on normal distribution) says they should? Or vice versa, would normally distributed wins yield bigger downswings than reality?

I take from your answer that as long as the sample statistics are accurate, assuming normal distribution is good enough and should be very close to reality after not all that much time.

The result of individual hands are actually Bernoulli distributed (at least if you were to get all the money in pre-flop and look at raw equities). The Central Limit Theorem states the sum of a large number of discrete random variables approaches the normal distribution. So the normal distribution becomes an excellent approximation for poker results over large numbers of hands. Live play results should be normal distributed, but the standard deviation and mean are both likely larger than online, so it may take more hands to look like a normal distribution. Also, live play is much slower, so it could take a lot of time before the results can really be well approximated by a normal distribution.

One of the well-studied bounds for the distance between the actual distribution of the scaled mean and the standard normal distribution is the Berry-Esseen inequality.

The absolute values of cash game winnings typically have a low skew (losing or winning one stack is quite common but winning more than a few starting stacks is rather uncommon), hence the B-E bound is quite tight even for small samples (hundreds of hands) and the distribution of the mean of the outcomes is close to normal.

On the contrary, the mean of the results of online jackpot SNGs is usually noticeably different from normal - both the negative and the positive 'tails' are 'fatter' - even for a sample of thousands of games because the outcome of a single tourney has a large positive skew (only 1 BI can be lost but thousands of BIs can be won in a single game). In other words, very hot or rather cold runs (depending on whether one hits the main jackpot soon enough) are more common than in a cash game or non-jackpot SNGs of the same EV and variance but neutral runs are less common. This can be verified by the SwongSim tool.

I'm not sure if this is useful but a few years ago I tried simulating tournament results with my own finish distribution for 180 seat tournaments (old - 18 paid).
Finish Dist in %. (0.712, 0.821, 0.876, 0.986, 0.821, 0.67, 1.041, 0.438, 0.602, (10 to 18) 0.575 each)

I did 500,000 simulations at each amount of games using the above finish position probabilities and plotted the the results. The right hand plot is a P-P plot of normal vs the simulated, the closer to a 45% straight line the closer the data to normal.

Even with the very spiked results of tournaments this simulation shows the results seem to come from a normal shaped distribution by the time I played about 100 games, and just about exact by 1000 games.

That's interesting. You start with a great deal of kurtosis, and it looks more like a Taleb distribution than a normal distribution, until you get 100+ games.