{"ID":2826105,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.19063","arxiv_id":"2512.19063","title":"Sharp Decoupling Inequalities for the Variances and Second Moments of Sums of Dependent Random Variables","abstract":"Both complete decoupling and tangent decoupling are classical tools aiming to compare two random processes where one has a weaker dependence structure. We give a new proof for the complete decoupling inequality, which provides a lower bound for the sum of dependent square-integrable nonnegative random variables $\\sum\\limits^n_{i=1} d_i$ \\[ \\frac{1}{2} \\mathbb E \\left( \\sum\\limits^n_{i=1} z_i \\right)^2 \\leq \\mathbb E \\left( \\sum\\limits^n_{i=1} d_i \\right)^2, \\] where $z_i \\stackrel{\\mathcal{L}}{=} d_i$ for all $i\\leq n$ and $z_i$'s are mutually independent. We will then provide the following sharp tangent decoupling inequalities \\[\\mathbb Var \\left( \\sum\\limits^n_{i=1} d_i\\right) \\leq 2 \\mathbb Var \\left( \\sum\\limits^n_{i=1} e_i\\right),\\] and \\[\\mathbb E \\left( \\sum\\limits^n_{i=1} d_i\\right)^2 \\leq 2 \\mathbb E \\left( \\sum\\limits^n_{i=1} e_i\\right)^2 - \\left[ \\mathbb E \\left( \\sum\\limits^n_{i=1} e_i\\right) \\right]^2,\\] where $\\{e_i\\}$ is the decoupled sequences of $\\{d_i\\}$ and $d_i$'s are not forced to be nonnegative. Applications to construct Chebyshev-type inequality and Paley-Zygmund-type inequality, and to bound the second moments of randomly stopped sums will be provided.","short_abstract":"Both complete decoupling and tangent decoupling are classical tools aiming to compare two random processes where one has a weaker dependence structure. We give a new proof for the complete decoupling inequality, which provides a lower bound for the sum of dependent square-integrable nonnegative random variables $\\sum\\l...","url_abs":"https://arxiv.org/abs/2512.19063","url_pdf":"https://arxiv.org/pdf/2512.19063v1","authors":"[\"Victor H. de la Pena\",\"Heyuan Yao\",\"Demissie Alemayehu\"]","published":"2025-12-22T06:01:17Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}