 Expectation closure error for a nonlinear recurrence

ID: expectation-closure-error-for-a-nonlinear-recurrence

A bound

E [B_{j + 1} ∣ B_{j}] \leq f (B_{j})

implies

E B_{j + 1} \leq E f (B_{j})

. It does not by itself imply

E B_{j + 1} \leq f (E B_{j})

. For

f (b) = b^{2} / n

, Jensen's inequality goes in the opposite direction. The missing contribution is the variance:

E [B_{j}^{2}] = (E B_{j})^{2} + Var (B_{j})

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