How relevant is historical performance for Elo rating? How far back are results of matches relevant for Elo purposes? If a player wins a few early matches then they will earn less points for later wins that if they'd lost those early matches. In the end there comes a point at which older results simply don't affect today's rating.
Matches with higher K factor will erase earlier history faster. I think that K factor has a linear effect on change in rating, so for the purpose of this calculation we can just assume constant K.
Let's say a FIDE player is rated 1800, and has k=20. Every month this player plays in one 4-game tournament, in which they play against opponents all rated exactly 2000, and scores exactly 50%. How fast does the rating converge?
After six months, they would be rated 1898. After one year, the player's rating would be 1949. After 18 months,the rating would be 1975, and after two years, it would be 1987. So it seems that the player is getting approximately halfway to their performance rating every six months or so in this scenario.
After 32 months, the player would be rated 1997, and after that the rating no longer moves because a rating difference of 3 has an expected score of 50% according to the table FIDE uses.