I guess it works both ways
Not Exactly: Correspondence, Classical, And Rapid Help In All Time-Controls, As - By Calculating Different Variations - You Will Gain A Better Understanding Of What Makes "Good-Moves" In Chess, So You Will Be Able To Make Good-Moves Faster In Bullet; Blitz Helps In Blitz, As Blitz Is Too Fast To Help In Rapid And Classical, And Too Slow To Help In Bullet; Bullet Helps In Bullet, As Bullet Is Too Fast To Help In Blitz, Rapid, Correspondence, And Classical.
Not necessarily true, at least in my own experience.
Time trouble usually comes about as you attempt to play good moves in faster controls. Once that kicks in, errors start to enter the game under time pressure. An empire of good moves built would have been ruined just because of these errors. In general, players who play rapid alone are often worse than players who play a mix of both if their long-control ratings are comparable.
More often than not, this is what I noticed when people face time trouble. And I believe this is what leads to Marcyful starting to incorporate bullet games as part of routine training.
I guess it works both ways
Not Exactly: Correspondence, Classical, And Rapid Help In All Time-Controls, As - By Calculating Different Variations - You Will Gain A Better Understanding Of What Makes "Good-Moves" In Chess, So You Will Be Able To Make Good-Moves Faster In Bullet; Blitz Helps In Blitz, As Blitz Is Too Fast To Help In Rapid, Correspondence, And Classical, And Too Slow To Help In Bullet; Bullet Helps In Bullet, As Bullet Is Too Fast To Help In Blitz, Rapid, Correspondence, And Classical.