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Sharp Threshold for the Convergence of Nonstationary Averaging

Saba Lepsveridze, Elchanan Mossel

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

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 . 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 . We show that the sequence necessarily converges when , while the convergence can fail. Second, we study complementary fixed shape regime, when is obtained by a fixed limiting density on . We show that under mild regularity assumptions, the sequence converges.
Paper Structure (17 sections, 16 theorems, 113 equations)

This paper contains 17 sections, 16 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Split trees and tagged lineages.
  4. Self-averaging sequences
  5. Renewal theory
  6. Consensus/social learning and stochastic approximation.
  7. Acknowledgements
  8. Reduction
  9. Majorization
  10. Main Technical Tools
  11. Tools for Upper Bounds
  12. Tools for Lower Bounds
  13. Proof of the Main Theorem
  14. Fixed Shape Analysis
  15. A Renewal Lemma for Multiplicative Recursions
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $f(n) = A (\log n)^{-\alpha}$ and $c(n) = B(\log n)^{\beta}$ with $\alpha, \beta > 0$.

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2: Informal
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2: Majorization
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • ...and 31 more