Robbins-Siegmund Theorem

The Robbins-Siegmund Theorem is a mathematical result that ensures, under certain conditions, a sequence of random variables stabilizes over time. Specifically, if each step’s expected value decreases or stabilizes thanks to some controlled factors—like a "learning rate"—then the sequence will almost surely converge to a finite limit. In simpler terms, it guarantees that processes, such as adaptive algorithms or estimates improving over time, won’t diverge wildly but will settle down eventually, making the behavior predictable and stable despite randomness.