Before revision:  (Recently, Wager and Xu (arXiv:2101.09855) independently proposed this regime and developed similar SDE and random ODE approximations  .) Our weak convergence theory covers both the classical multi-armed and linear bandit settings,  and can be used, for instance, to obtain insight about the characteristics of the regret distribution when there is information sharing among arms, as well as the effects of variance estimation, model mis-specification and batched updates in bandit learning.
After revision:  (Recently, Wager and Xu (arXiv:2101.09855) independently proposed this regime and developed similar SDE and random ODE approximations .) Our weak convergence theory covers both the classical multi-armed and linear bandit settings, is developed from first principles using the Continuous Mapping Theorem and can be used, for instance, to obtain insight about the characteristics of the regret distribution when there is information sharing among arms, as well as the effects of variance estimation, model mis-specification and batched updates in bandit learning.