Sharp Threshold for the Convergence of Nonstationary Averaging

Abstract

We study non-stationary averaging processes, where each term of a sequence is a weighted average of previous terms, namely $a_{n+1}={\sum }_{j=1}^n{p}_n(j)a_j$. Our results extend classical theory in two distinct regimes. First, we prove a sharp threshold for convergence in the regime where the weights are bounded between two envelopes $(\log n{)}^{-\hat{I}\pm }\le n{p}_n(\cdot )\le (\log n{)}^2$. We show that the sequence necessarily converges when $\hat{I}\pm +{\hat{I}}^2/2\le 1$, while $\hat{I}\pm +{\hat{I}}^2/2>1$ the convergence can fail. Second, we study complementary fixed shape regime, when $p_n$ is obtained by a fixed limiting density on $(0,1)$. We show that under mild regularity assumptions, the sequence converges.

29 pages