Consider a sequence of zeros and ones which is generated as follows. At each t we compute p(t) which is the fraction of ones in the sequence up until now. Then we choose zero with probability p and one probably one minus p. It seems obvious that p(t) will approach 1/2. Is there a simple method to prove convergence for this and similar adaptive processes?
2026-04-01 00:22:42.1775002962
Convergence in probability of adaptive process
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In case anyone has the same question, the Robbins-Siegmund Theorem makes this easy to prove.
Robbins, Herbert, and David Siegmund. "A convergence theorem for non negative almost supermartingales and some applications." Optimizing methods in statistics. Academic Press, 1971. 233-257.