Sharp Threshold for the Convergence of Nonstationary Averaging

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.