I was reading the Wikipedia article about the Hot Hand Fallacy, and I decided to try a little bit of statistical analysis myself. I've always found the idea of a player being hot or cold a bit suspicious. On one hand, it kind of feels right. Sometimes a player just has it, right? And there's no reason events in a game need to be considered completely independent. On the other hand, maybe the phenomenon can be explained by our natural inclination to look for patterns.
I chose baseball to look at. I downloaded the 2015 event database from Retrosheet, and wrote some code to calculate autocorrelation functions. I generated a long list of every plate appearance outcome for the whole season. The list is grouped by player and in order for each player. I labeled all hits as 1 and all outs as 0. I then computed the autocorrelation of this sequence.
I inserted the black line to guide the eye. As you can see, there is some correlation between successful plate appearances. I think this suggests that the time-scale for hotness or coldness is about ten plate appearances. Two games, or so.
Also, it isn't a huge effect. It looks like 5% or something.