Sharp Threshold for the Convergence of Nonstationary Averaging

TL;DR

This paper establishes a precise threshold for convergence in non-stationary averaging processes, extending classical results and analyzing two regimes: bounded weights and fixed shape densities.

Contribution

It provides a sharp convergence threshold for non-stationary averaging sequences with bounded weights and proves convergence under fixed shape density conditions.

Findings

Convergence occurs if  + eta/2  1 in bounded weight regime.

Sequences may fail to converge if  + eta/2 > 1.

Sequences with fixed limiting density on (0,1) converge under mild conditions.

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 $(\log n)^{-\alpha} \le np_n(\cdot) \leq (\log n)^{\beta}$. We show that the sequence necessarily converges when $\alpha + \beta / 2 \leq 1$, while $\alpha + \beta / 2 > 1$ 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.