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Rate of convergence of empirical measures for exchangeable sequences

In: Mathematica Slovaca, vol. 67, no. 6
Patrizia Berti - Luca Pratelli - Pietro Rigo
Detaily:
Rok, strany: 2017, 1557 - 1570
Kľúčové slová:
empirical measure, exchangeability, predictive measure, random probability measure, rate of convergence
O článku:
Let $S$ be a finite set, $(Xn)$ an exchangeable sequence of $S$-valued random variables, and $μn=(1/n) ∑i=1nδXi$ the empirical measure. Then, $μn(B)\overset{a.s.}\longrightarrow μ(B)$ for all $B\subset S$ and some (essentially unique) random probability measure $μ$. Denote by $\mathcal{L}(Z)$ the probability distribution of any random variable $Z$. Under some assumptions on $\mathcal{L}(μ)$, it is shown that \begin{equation*} ((a) / (n))≤ρ[\mathcal{L}(μn), \mathcal{L}(μ)]≤((b) / (n))  and  ρ[\mathcal{L}(μn), \mathcal{L}(an)]≤((c) / (nu)) \end{equation*} where $ρ$ is the bounded Lipschitz metric and $an(·)=P(Xn+1\in·\mid X1,…,Xn)$ is the predictive measure. The constants $a, b, c>0$ and $u\in (((1) / (2)), 1]$ depend on $\mathcal{L}(μ)$ and card$(S)$ only.
Ako citovať:
ISO 690:
Berti, P., Pratelli, L., Rigo, P. 2017. Rate of convergence of empirical measures for exchangeable sequences. In Mathematica Slovaca, vol. 67, no.6, pp. 1557-1570. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0070

APA:
Berti, P., Pratelli, L., Rigo, P. (2017). Rate of convergence of empirical measures for exchangeable sequences. Mathematica Slovaca, 67(6), 1557-1570. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0070
O vydaní:
Publikované: 27. 11. 2017