How Historical Noise Is Attenuated
Series: Stochastic Approximation from First Principles · Part 20 of 22 The expectation can approach zero while the error remains widely dispersed. To determine whether noisy trajectories also concentrate, analyze: $$ \operatorname{Var}(\Delta_k). $$Isolate the Random Part Write the explicit recursion as: $$ \Delta_k=A_k+B_k, $$where: $$ A_k = \left[ \prod_{i=0}^{k-1}(1-\alpha_i) \right]\Delta_0 $$is deterministic, and: $$ B_k = \sum_{j=0}^{k-1}w_{j,k}\eta_j, $$with: $$ w_{j,k} = \left[ \prod_{i=j+1}^{k-1}(1-\alpha_i) \right]\beta_j. $$Because $A_k$ is deterministic: $$ \operatorname{Var}(\Delta_k)=\operatorname{Var}(B_k). $$If the noise terms are independent or at least uncorrelated: ...