On Complete Convergence in Mean of Normed Sums of Independent Random Elements in Banach Spaces
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Bibliographic record
Abstract
Abstract For a sequence of random elements {T n , n ≥ 1} in a real separable Banach space 𝒳, we study the notion of T n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T n converging completely to 0 and (ii) T n converging to 0 in mean of order p. When 𝒳 is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements {V n , n ≥ 1} in 𝒳 and a sequence of constants b n → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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