Why does the elephant random walk behave as the randomized play-the-winner rule?

The randomized play-the-winner rule (RPW) is a response-adaptive design proposed by Wei and Durham (1978) for sequentially randomizing patients to treatments in a two-treatment clinical trial so that more patients are assigned to the better treatment as the clinical trial goes on. The elephant random walk (ERW) proposed by Schutz and Trimper (2004) is a non-Markovian discrete-time random walk on Z which has a link to a famous saying that elephants can always remember where they have been. The asymptotic behaviors of RPW rule and ERW have been studied in litterateurs independently, and their asymptotic behaviors are very similar. In this paper, we show that the RPW rule is a biased ERW and link them with the recursive stochastic algorithm. With the help of a recursive stochastic algorithm, we obtain the strong invariance principle of the ERW as well as multi-dimensional ERW with random step sizes. By the strong invariance principle, the central limit theorem and precise law of the iterated logarithm are obtained for both the multi-dimensional ERW, the multi-dimensional ERW with random step sizes and their centers of mass.