{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:59:25Z","timestamp":1787324365743,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>Almost sure convergence rates for linear algorithms $h_{k+1} = h_k +\\frac{1}{k^\\chi} (b_k-A_kh_k)$ are studied, where $\\chi\\in(0,1)$, $\\{A_{k}\\}_{k=1}^\\infty$ are symmetric, positive semidefinite random matrices and $\\{b_{k}\\}_{k=1}^\\infty$ are random vectors. It is shown that $n^{\\gamma}|h_n- A^{-1}b|\\rightarrow 0 \\,$ a.s. for the $\\gamma\\in[0,\\chi)$, positive definite $A$ and vector $b$ such that $\\frac{1}{n^{\\chi-\\gamma}}\\sum_{k=1}^n (A_{k}- A)\\to 0$ and $\\frac{1}{n^{\\chi-\\gamma}}\\sum_{k=1}^n (b_k-b)\\to 0$ a.s. When $\\chi-\\gamma\\in\\left(\\frac12,1\\right)$, these assumptions are implied by the Marcinkiewicz strong law of large numbers, which allows the $\\{A_k\\}$ and $\\{b_k\\}$ to have heavy tails, long-range dependence, or both. Finally, corroborating experimental outcomes and decreasing-gain design considerations are provided.<\/jats:p>","DOI":"10.1137\/14095707x","type":"journal-article","created":{"date-parts":[[2015,6,9]],"date-time":"2015-06-09T11:43:50Z","timestamp":1433850230000},"page":"1484-1508","source":"Crossref","is-referenced-by-count":5,"title":["Convergence Rates and Decoupling in Linear Stochastic Approximation Algorithms"],"prefix":"10.1137","volume":"53","author":[{"given":"Michael A.","family":"Kouritzin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Samira","family":"Sadeghi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,6,9]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"A. Benveniste, M. Me\u0301tivier, and P. 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