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Publish exact native sequential mean testing reproduction

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\begin{thebibliography}{26}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
  \providecommand{\doi}[1]{doi: #1}\else
  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi

\bibitem[Agrawal(2023)]{agrawal2022bandits}
Agrawal, S.
\newblock \emph{Bandits with Heavy Tails: Algorithms Analysis and Optimality}.
\newblock PhD thesis, Tata Institute of Fundamental Research, 2023.
\newblock URL \url{http://hdl.handle.net/10603/478863}.

\bibitem[Agrawal \& Ramdas(2025)Agrawal and Ramdas]{agrawal2025stopping}
Agrawal, S. and Ramdas, A.
\newblock On stopping times of power-one sequential tests: Tight lower and upper bounds.
\newblock \emph{arXiv preprint arXiv:2504.19952}, 2025.

\bibitem[Agrawal et~al.(2020)Agrawal, Juneja, and Glynn]{agrawal2020optimal}
Agrawal, S., Juneja, S., and Glynn, P.
\newblock Optimal $\delta$-correct best-arm selection for heavy-tailed distributions.
\newblock In \emph{Algorithmic Learning Theory}, pp.\  61--110. PMLR, 2020.

\bibitem[Agrawal et~al.(2021{\natexlab{a}})Agrawal, Juneja, and Koolen]{pmlr-v134-agrawal21a}
Agrawal, S., Juneja, S.~K., and Koolen, W.~M.
\newblock Regret minimization in heavy-tailed bandits.
\newblock In \emph{Proceedings of Thirty Fourth Conference on Learning Theory}, volume 134 of \emph{Proceedings of Machine Learning Research}, pp.\  26--62. PMLR, 15--19 Aug 2021{\natexlab{a}}.

\bibitem[Agrawal et~al.(2021{\natexlab{b}})Agrawal, Koolen, and Juneja]{agrawal2021optimal}
Agrawal, S., Koolen, W.~M., and Juneja, S.
\newblock Optimal best-arm identification methods for tail-risk measures.
\newblock \emph{Advances in Neural Information Processing Systems}, 34:\penalty0 25578--25590, 2021{\natexlab{b}}.

\bibitem[Anscombe(1952)]{anscombe1952large}
Anscombe, F.~J.
\newblock Large-sample theory of sequential estimation.
\newblock In \emph{Mathematical Proceedings of the Cambridge Philosophical Society}, volume~48, pp.\  600--607. Cambridge University Press, 1952.

\bibitem[Asmussen(2003)]{asmussen2003applied}
Asmussen, S.
\newblock \emph{Applied probability and queues}.
\newblock Springer, 2003.

\bibitem[Billingsley(2017)]{billingsley2017probability}
Billingsley, P.
\newblock \emph{Probability and measure}.
\newblock John Wiley \& Sons, 2017.

\bibitem[Burnetas \& Katehakis(1996)Burnetas and Katehakis]{burnetas1996optimal}
Burnetas, A.~N. and Katehakis, M.~N.
\newblock Optimal adaptive policies for sequential allocation problems.
\newblock \emph{Advances in Applied Mathematics}, 17\penalty0 (2):\penalty0 122--142, 1996.

\bibitem[Chernoff(1959)]{chernoff1992sequential}
Chernoff, H.
\newblock Sequential design of experiments.
\newblock \emph{Ann. Math. Statist.}, 30\penalty0 (4):\penalty0 755--770, 1959.

\bibitem[Darling \& Robbins(1967)Darling and Robbins]{darling1967iterated}
Darling, D.~A. and Robbins, H.
\newblock Iterated logarithm inequalities.
\newblock \emph{Proceedings of the National Academy of Sciences}, 57\penalty0 (5):\penalty0 1188--1192, 1967.

\bibitem[Deep et~al.(2024)Deep, Bassamboo, and Juneja]{deep2024asymptotically}
Deep, V., Bassamboo, A., and Juneja, S.~K.
\newblock Asymptotically optimal and computationally efficient average treatment effect estimation in a/b testing.
\newblock 2024.

\bibitem[Deep et~al.(2025)Deep, Bassamboo, and Juneja]{deep2025asymptotic}
Deep, V., Bassamboo, A., and Juneja, S.
\newblock Asymptotic optimality theory of confidence intervals of the mean.
\newblock \emph{arXiv preprint arXiv:2501.19126}, 2025.

\bibitem[Fan \& Glynn(2025)Fan and Glynn]{fan2025fragility}
Fan, L. and Glynn, P.~W.
\newblock The fragility of optimized bandit algorithms.
\newblock \emph{Operations Research}, 73\penalty0 (6):\penalty0 3173--3198, 2025.

\bibitem[Gut(2009)]{gut2009stopped}
Gut, A.
\newblock \emph{Stopped random walks}.
\newblock Springer, 2009.

\bibitem[Honda \& Takemura(2010)Honda and Takemura]{honda2010asymptotically}
Honda, J. and Takemura, A.
\newblock An asymptotically optimal bandit algorithm for bounded support models.
\newblock In \emph{COLT}, pp.\  67--79, 2010.

\bibitem[Honda \& Takemura(2015)Honda and Takemura]{honda2015non}
Honda, J. and Takemura, A.
\newblock Non-asymptotic analysis of a new bandit algorithm for semi-bounded rewards.
\newblock \emph{J. Mach. Learn. Res.}, 16:\penalty0 3721--3756, 2015.

\bibitem[Jourdan et~al.(2022)Jourdan, Degenne, Baudry, de~Heide, and Kaufmann]{jourdan2022top}
Jourdan, M., Degenne, R., Baudry, D., de~Heide, R., and Kaufmann, E.
\newblock Top two algorithms revisited.
\newblock \emph{Advances in Neural Information Processing Systems}, 35:\penalty0 26791--26803, 2022.

\bibitem[Lai \& Robbins(1985)Lai and Robbins]{lai1985asymptotically}
Lai, T.~L. and Robbins, H.
\newblock Asymptotically efficient adaptive allocation rules.
\newblock \emph{Advances in Applied Mathematics}, 6\penalty0 (1):\penalty0 4--22, 1985.

\bibitem[Mukhopadhyay(2020)]{mukhopadhyay2020asymptotic}
Mukhopadhyay, N.
\newblock Asymptotic normality of sequential stopping times with applications: Confidence intervals for an exponential mean.
\newblock \emph{Calcutta Statistical Association Bulletin}, 72\penalty0 (1):\penalty0 17--34, 2020.

\bibitem[Panda \& Agrawal(2026)Panda and Agrawal]{panda2026regret}
Panda, S. and Agrawal, S.
\newblock Regret tail characterization of optimal bandit algorithms with generic rewards.
\newblock \emph{arXiv preprint arXiv:2604.14876}, 2026.

\bibitem[Robbins \& Siegmund(1974)Robbins and Siegmund]{robbins1974expected}
Robbins, H. and Siegmund, D.
\newblock The expected sample size of some tests of power one.
\newblock \emph{The Annals of Statistics}, 2\penalty0 (3):\penalty0 415--436, 1974.

\bibitem[Siegmund(2013)]{siegmund2013sequential}
Siegmund, D.
\newblock \emph{Sequential Analysis: Tests and Confidence Intervals}.
\newblock Springer Science \& Business Media, 2013.

\bibitem[Wald(1992)]{wald1992sequential}
Wald, A.
\newblock Sequential tests of statistical hypotheses.
\newblock In \emph{Breakthroughs in statistics: Foundations and basic theory}, pp.\  256--298. Springer, 1992.

\bibitem[Wald \& Wolfowitz(1948)Wald and Wolfowitz]{wald1948optimum}
Wald, A. and Wolfowitz, J.
\newblock Optimum character of the sequential probability ratio test.
\newblock \emph{The Annals of Mathematical Statistics}, pp.\  326--339, 1948.

\bibitem[Wang et~al.(2026)Wang, Agrawal, and Ramdas]{wang2026almost}
Wang, H., Agrawal, S., and Ramdas, A.
\newblock Almost sure null bankruptcy of testing-by-betting strategies.
\newblock \emph{arXiv preprint arXiv:2602.08888}, 2026.

\end{thebibliography}

	\begin{thebibliography}{26}
	\providecommand{\natexlab}[1]{#1}
	\providecommand{\url}[1]{\texttt{#1}}
	\expandafter\ifx\csname urlstyle\endcsname\relax
	\providecommand{\doi}[1]{doi: #1}\else
	\providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi

	\bibitem[Agrawal(2023)]{agrawal2022bandits}
	Agrawal, S.
	\newblock \emph{Bandits with Heavy Tails: Algorithms Analysis and Optimality}.
	\newblock PhD thesis, Tata Institute of Fundamental Research, 2023.
	\newblock URL \url{http://hdl.handle.net/10603/478863}.

	\bibitem[Agrawal \& Ramdas(2025)Agrawal and Ramdas]{agrawal2025stopping}
	Agrawal, S. and Ramdas, A.
	\newblock On stopping times of power-one sequential tests: Tight lower and upper bounds.
	\newblock \emph{arXiv preprint arXiv:2504.19952}, 2025.

	\bibitem[Agrawal et~al.(2020)Agrawal, Juneja, and Glynn]{agrawal2020optimal}
	Agrawal, S., Juneja, S., and Glynn, P.
	\newblock Optimal $\delta$-correct best-arm selection for heavy-tailed distributions.
	\newblock In \emph{Algorithmic Learning Theory}, pp.\ 61--110. PMLR, 2020.

	\bibitem[Agrawal et~al.(2021{\natexlab{a}})Agrawal, Juneja, and Koolen]{pmlr-v134-agrawal21a}
	Agrawal, S., Juneja, S.~K., and Koolen, W.~M.
	\newblock Regret minimization in heavy-tailed bandits.
	\newblock In \emph{Proceedings of Thirty Fourth Conference on Learning Theory}, volume 134 of \emph{Proceedings of Machine Learning Research}, pp.\ 26--62. PMLR, 15--19 Aug 2021{\natexlab{a}}.

	\bibitem[Agrawal et~al.(2021{\natexlab{b}})Agrawal, Koolen, and Juneja]{agrawal2021optimal}
	Agrawal, S., Koolen, W.~M., and Juneja, S.
	\newblock Optimal best-arm identification methods for tail-risk measures.
	\newblock \emph{Advances in Neural Information Processing Systems}, 34:\penalty0 25578--25590, 2021{\natexlab{b}}.

	\bibitem[Anscombe(1952)]{anscombe1952large}
	Anscombe, F.~J.
	\newblock Large-sample theory of sequential estimation.
	\newblock In \emph{Mathematical Proceedings of the Cambridge Philosophical Society}, volume~48, pp.\ 600--607. Cambridge University Press, 1952.

	\bibitem[Asmussen(2003)]{asmussen2003applied}
	Asmussen, S.
	\newblock \emph{Applied probability and queues}.
	\newblock Springer, 2003.

	\bibitem[Billingsley(2017)]{billingsley2017probability}
	Billingsley, P.
	\newblock \emph{Probability and measure}.
	\newblock John Wiley \& Sons, 2017.

	\bibitem[Burnetas \& Katehakis(1996)Burnetas and Katehakis]{burnetas1996optimal}
	Burnetas, A.~N. and Katehakis, M.~N.
	\newblock Optimal adaptive policies for sequential allocation problems.
	\newblock \emph{Advances in Applied Mathematics}, 17\penalty0 (2):\penalty0 122--142, 1996.

	\bibitem[Chernoff(1959)]{chernoff1992sequential}
	Chernoff, H.
	\newblock Sequential design of experiments.
	\newblock \emph{Ann. Math. Statist.}, 30\penalty0 (4):\penalty0 755--770, 1959.

	\bibitem[Darling \& Robbins(1967)Darling and Robbins]{darling1967iterated}
	Darling, D.~A. and Robbins, H.
	\newblock Iterated logarithm inequalities.
	\newblock \emph{Proceedings of the National Academy of Sciences}, 57\penalty0 (5):\penalty0 1188--1192, 1967.

	\bibitem[Deep et~al.(2024)Deep, Bassamboo, and Juneja]{deep2024asymptotically}
	Deep, V., Bassamboo, A., and Juneja, S.~K.
	\newblock Asymptotically optimal and computationally efficient average treatment effect estimation in a/b testing.
	\newblock 2024.

	\bibitem[Deep et~al.(2025)Deep, Bassamboo, and Juneja]{deep2025asymptotic}
	Deep, V., Bassamboo, A., and Juneja, S.
	\newblock Asymptotic optimality theory of confidence intervals of the mean.
	\newblock \emph{arXiv preprint arXiv:2501.19126}, 2025.

	\bibitem[Fan \& Glynn(2025)Fan and Glynn]{fan2025fragility}
	Fan, L. and Glynn, P.~W.
	\newblock The fragility of optimized bandit algorithms.
	\newblock \emph{Operations Research}, 73\penalty0 (6):\penalty0 3173--3198, 2025.

	\bibitem[Gut(2009)]{gut2009stopped}
	Gut, A.
	\newblock \emph{Stopped random walks}.
	\newblock Springer, 2009.

	\bibitem[Honda \& Takemura(2010)Honda and Takemura]{honda2010asymptotically}
	Honda, J. and Takemura, A.
	\newblock An asymptotically optimal bandit algorithm for bounded support models.
	\newblock In \emph{COLT}, pp.\ 67--79, 2010.

	\bibitem[Honda \& Takemura(2015)Honda and Takemura]{honda2015non}
	Honda, J. and Takemura, A.
	\newblock Non-asymptotic analysis of a new bandit algorithm for semi-bounded rewards.
	\newblock \emph{J. Mach. Learn. Res.}, 16:\penalty0 3721--3756, 2015.

	\bibitem[Jourdan et~al.(2022)Jourdan, Degenne, Baudry, de~Heide, and Kaufmann]{jourdan2022top}
	Jourdan, M., Degenne, R., Baudry, D., de~Heide, R., and Kaufmann, E.
	\newblock Top two algorithms revisited.
	\newblock \emph{Advances in Neural Information Processing Systems}, 35:\penalty0 26791--26803, 2022.

	\bibitem[Lai \& Robbins(1985)Lai and Robbins]{lai1985asymptotically}
	Lai, T.~L. and Robbins, H.
	\newblock Asymptotically efficient adaptive allocation rules.
	\newblock \emph{Advances in Applied Mathematics}, 6\penalty0 (1):\penalty0 4--22, 1985.

	\bibitem[Mukhopadhyay(2020)]{mukhopadhyay2020asymptotic}
	Mukhopadhyay, N.
	\newblock Asymptotic normality of sequential stopping times with applications: Confidence intervals for an exponential mean.
	\newblock \emph{Calcutta Statistical Association Bulletin}, 72\penalty0 (1):\penalty0 17--34, 2020.

	\bibitem[Panda \& Agrawal(2026)Panda and Agrawal]{panda2026regret}
	Panda, S. and Agrawal, S.
	\newblock Regret tail characterization of optimal bandit algorithms with generic rewards.
	\newblock \emph{arXiv preprint arXiv:2604.14876}, 2026.

	\bibitem[Robbins \& Siegmund(1974)Robbins and Siegmund]{robbins1974expected}
	Robbins, H. and Siegmund, D.
	\newblock The expected sample size of some tests of power one.
	\newblock \emph{The Annals of Statistics}, 2\penalty0 (3):\penalty0 415--436, 1974.

	\bibitem[Siegmund(2013)]{siegmund2013sequential}
	Siegmund, D.
	\newblock \emph{Sequential Analysis: Tests and Confidence Intervals}.
	\newblock Springer Science \& Business Media, 2013.

	\bibitem[Wald(1992)]{wald1992sequential}
	Wald, A.
	\newblock Sequential tests of statistical hypotheses.
	\newblock In \emph{Breakthroughs in statistics: Foundations and basic theory}, pp.\ 256--298. Springer, 1992.

	\bibitem[Wald \& Wolfowitz(1948)Wald and Wolfowitz]{wald1948optimum}
	Wald, A. and Wolfowitz, J.
	\newblock Optimum character of the sequential probability ratio test.
	\newblock \emph{The Annals of Mathematical Statistics}, pp.\ 326--339, 1948.

	\bibitem[Wang et~al.(2026)Wang, Agrawal, and Ramdas]{wang2026almost}
	Wang, H., Agrawal, S., and Ramdas, A.
	\newblock Almost sure null bankruptcy of testing-by-betting strategies.
	\newblock \emph{arXiv preprint arXiv:2602.08888}, 2026.

	\end{thebibliography}