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We draw connections between the Cram\\'er-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL), and the sequential probability ratio test -- our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes, and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.","url_abs":"https://arxiv.org/abs/1810.08240v9","url_pdf":"https://arxiv.org/pdf/1810.08240v9.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"uniform-nonparametric-non-asymptotic","repo_url":"https://github.com/gostevehoward/confseq","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"uniform-nonparametric-non-asymptotic","repo_url":"https://github.com/gostevehoward/cspaper","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"uniform-nonparametric-non-asymptotic","repo_url":"https://github.com/gsbDBI/adaptive-confidence-intervals","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}},{"paper_slug":"uniform-nonparametric-non-asymptotic","repo_url":"https://github.com/wannabesmith/confseq","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1810.08240","atlas_url":"https://app.syntology.ai/?focus=1810.08240","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1810.08240"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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