TY - JOUR

T1 - Existence of an infinite particle limit of stochastic ranking process

AU - Hattori, Kumiko

AU - Hattori, Tetsuya

N1 - Funding Information:
The research of K. Hattori is supported in part by a Grant-in-Aid for Scientific Research (C) 16540101 from the Ministry of Education, Culture, Sports, Science and Technology, and the research of T. Hattori is supported in part by a Grant-in-Aid for Scientific Research (B) 17340022 from the Ministry of Education, Culture, Sports, Science and Technology.

PY - 2009/3

Y1 - 2009/3

N2 - We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon.co.jp). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time-dependent distribution. A core of the proof is the law of large numbers for dependent random variables.

AB - We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon.co.jp). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time-dependent distribution. A core of the proof is the law of large numbers for dependent random variables.

KW - Dependent random variables

KW - Hydrodynamic limit

KW - Law of large numbers

KW - Stochastic ranking process

UR - http://www.scopus.com/inward/record.url?scp=60549089835&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60549089835&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2008.05.006

DO - 10.1016/j.spa.2008.05.006

M3 - Article

AN - SCOPUS:60549089835

VL - 119

SP - 966

EP - 979

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 3

ER -