A WEAR LAW OF LARGE NUMBERS FOR EMPIRICAL MEASURES VIA STEINS METHOD

• 成果类型: Article
• 署名作者: REINERT, G
• 署名单位: University of Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988389
• 发表日期: 1995
• 页码: 334-354
• 关键词: poisson approximation; THEOREM

摘要

Let E be a locally compact Hausdorff space with countable basis and let (X(i))(i epsilon N) be a family of random elements on E with (1/n) Sigma(i=1)(n) L (X(i)) double right arrow(>)v mu(n --> infinity) for a measure mu with parallel to mu parallel to less than or equal to 1. Conditions are derived under which L ((1/n) Sigma(i=1)(n) (delta)X(i)) double right arrow(w) delta(mu)(n --> infinity), where delta(x), denotes the Dirac measure at x. The proof being based on Stein's method, there are generalisations that allow for weak dependence between the X(i)'s. As examples, a dissociated family and an immigration-death process are considered. The latter illustrates the possible applications in proving convergence of stochastic processes.