Expectation closure error for a nonlinear recurrence
= Expectation closure error for a nonlinear recurrence
A bound $\mathbb E[B_{j+1}\mid B_j]\leq f(B_j)$ implies $\mathbb E B_{j+1}\leq\mathbb E f(B_j)$. It does not by itself imply $\mathbb E B_{j+1}\leq f(\mathbb E B_j)$. For $f(b)=b^2/n$, <Jensen's inequality> goes in the opposite direction. The missing contribution is the <variance>: $\mathbb E[B_j^2]=(\mathbb E B_j)^2+\operatorname{Var}(B_j)$.