Sharp Threshold for the Convergence of Nonstationary Averaging
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908894224187392 |
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| author | Lepsveridze, Saba Mossel, Elchanan |
| author_facet | Lepsveridze, Saba Mossel, Elchanan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Threshold for the Convergence of Nonstationary Averaging Lepsveridze, Saba Mossel, Elchanan Probability 60K05, 60K25, 60F15, 40A05, G.2.0; G.3 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. |
| title | Sharp Threshold for the Convergence of Nonstationary Averaging |
| topic | Probability 60K05, 60K25, 60F15, 40A05, G.2.0; G.3 |
| url | https://arxiv.org/abs/2603.16678 |