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Balls-in-bins with feedback and Brownian Motion
| Content Provider | Semantic Scholar |
|---|---|
| Author | Oliveira, Roberto |
| Copyright Year | 2008 |
| Abstract | In a balls-in-bins process with feedback, balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = n p for p > 0, and our goal is to study the fine behavior of this process with two bins and a large initial number t of balls. Perhaps surprisingly, Brownian Motions are an essential part of both our proofs. For p > 1/2, it was known that with probability 1 one of the bins will lead the process at all large enough times. We show that if the first bin starts with t + λ √ t balls (for constant λ ∈ R), the probability that it always or eventually leads has a non-trivial limit depending on λ. For p ≤ 1/2, it was known that with probability 1 the bins will alternate in leadership. We show, however, that if the initial fraction of balls in one of the bins is > 1/2, the time until it is overtaken by the remaining bin scales like �(t 1+1/(1 2p) ) for p < 1/2 and exp(�(t)) for p = 1/2. In fact, the overtaking time has a non-trivial distribution around the scaling factors, which we determine explicitly. Our proofs use a continuous-time embedding of the balls-in-bins process (due to Rubin) and a non-standard approximation of the process by Brownian Motion. The techniques presented also extend to more general functions f. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/math/0510648v1.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0510648v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
