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<p>This work establishes complete convergence and complete <i>q</i>-th moment convergence for weighted sums of independent random elements taking values in Rademacher type <i>p</i> Banach spaces without the assumption of identical distribution. The results are derived under the moment condition <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {E} R(\Vert X\Vert )&lt;\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">E</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mo stretchy="false">‖</mo> <mi>X</mi> <mo stretchy="false">‖</mo> <mo stretchy="false">)</mo> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R(\cdot )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a regularly varying function. We also examine the converses of these results and provide three illustrative examples.</p>

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Complete and Complete q-th Moment Convergence for Weighted Sums in Banach Spaces

  • 期刊论文

  • 发表日期：

    2026年7月27日

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  • Nguyen Van Huan,
  • Nguyen Huu Hieu

摘要

This work establishes complete convergence and complete q-th moment convergence for weighted sums of independent random elements taking values in Rademacher type p Banach spaces without the assumption of identical distribution. The results are derived under the moment condition \(\mathbb {E} R(\Vert X\Vert )<\infty \) E R ( ‖ X ‖ ) < ∞ , where \(R(\cdot )\) R ( · ) is a regularly varying function. We also examine the converses of these results and provide three illustrative examples.