## The Annals of Applied Probability

### Central limit theorem for an adaptive randomly reinforced urn model

#### Abstract

The generalized Pólya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with a diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the urn model in practice. Specifically, we establish the law of large numbers and a central limit theorem for the number of sampled balls. To establish these results, we develop new techniques involving last exit times and crossing time analyses of the proportion of balls in the urn. To obtain precise estimates in these techniques, we establish results on the harmonic moments of the total number of balls in the urn. Finally, we describe our main results in the context of an application to response-adaptive randomization in clinical trials. Our simulation experiments in this context demonstrate the ease and scope of our model.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 2956-3003.

Dates
Revised: October 2016
First available in Project Euclid: 3 November 2017

https://projecteuclid.org/euclid.aoap/1509696039

Digital Object Identifier
doi:10.1214/16-AAP1274

Mathematical Reviews number (MathSciNet)
MR3719951

Zentralblatt MATH identifier
1379.60025

#### Citation

Ghiglietti, Andrea; Vidyashankar, Anand N.; Rosenberger, William F. Central limit theorem for an adaptive randomly reinforced urn model. Ann. Appl. Probab. 27 (2017), no. 5, 2956--3003. doi:10.1214/16-AAP1274. https://projecteuclid.org/euclid.aoap/1509696039

#### References

• [1] Aletti, G., Ghiglietti, A. and Paganoni, A. M. (2013). Randomly reinforced urn designs with prespecified allocations. J. Appl. Probab. 50 486–498.
• [2] Aletti, G., May, C. and Secchi, P. (2009). A central limit theorem, and related results, for a two-color randomly reinforced urn. Adv. in Appl. Probab. 41 829–844.
• [3] Aletti, G., May, C. and Secchi, P. (2012). A functional equation whose unknown is $P([0,1])$ valued. J. Theoret. Probab. 25 1207–1232.
• [4] Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39 1801–1817.
• [5] Bai, Z. D. and Hu, F. (1999). Asymptotic theorems for urn models with nonhomogeneous generating matrices. Stochastic Process. Appl. 80 87–101.
• [6] Bai, Z.-D. and Hu, F. (2005). Asymptotics in randomized URN models. Ann. Appl. Probab. 15 914–940.
• [7] Bai, Z. D., Hu, F. and Zhang, L.-X. (2002). Gaussian approximation theorems for urn models and their applications. Ann. Appl. Probab. 12 1149–1173.
• [8] Chauvin, B., Pouyanne, N. and Sahnoun, R. (2011). Limit distributions for large Pólya urns. Ann. Appl. Probab. 21 1–32.
• [9] Chen, L. H. Y. (1978). A short note on the conditional Borel–Cantelli lemma. Ann. Probab. 6 699–700.
• [10] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
• [11] Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a favourable response. Canad. J. Statist. 26 479–495.
• [12] Etemadi, N., Sriram, T. N. and Vidyashankar, A. N. (1997). $L_{p}$ convergence of reciprocals of sample means with applications to sequential estimation in linear regression. J. Statist. Plann. Inference 65 1–15.
• [13] Ghiglietti, A. and Paganoni, A. M. (2014). Statistical properties of two-color randomly reinforced urn design targeting fixed allocations. Electron. J. Stat. 8 708–737.
• [14] Ghiglietti, A. and Paganoni, A. M. (2016). An urn model to construct an efficient test procedure for response adaptive designs. Stat. Methods Appl. 25 211–226.
• [15] Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. Wiley, Hoboken, NJ.
• [16] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245.
• [17] Mahmoud, H. M. (2009). Pólya Urn Models. CRC Press, Boca Raton, FL.
• [18] Muliere, P., Paganoni, A. M. and Secchi, P. (2006). A randomly reinforced urn. J. Statist. Plann. Inference 136 1853–1874.
• [19] Rosenberger, W. F. (2002). Randomized urn models and sequential design. Sequential Anal. 21 1–41.
• [20] Rosenberger, W. F. and Lachin, J. M. (2002). Randomization in Clinical Trials: Theory and Practice. Wiley, New York.
• [21] Rosenberger, W. F., Stallard, N., Ivanova, A., Harper, C. N. and Ricks, M. L. (2001). Optimal adaptive designs for binary response trials. Biometrics 57 909–913.
• [22] Smythe, R. T. (1996). Central limit theorems for urn models. Stochastic Process. Appl. 65 115–137.
• [23] Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. J. Amer. Statist. Assoc. 73 840–843.
• [24] Zhang, L. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics 62 562–569.
• [25] Zhang, L.-X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Probab. 16 340–369.
• [26] Zhang, L.-X., Hu, F., Cheung, S. H. and Chan, W. S. (2011). Immigrated urn models—theoretical properties and applications. Ann. Statist. 39 643–671.