Let \$S\$ be a finite set, \$(X_n)\$ an exchangeable sequence of \$S\$-valued random variables, and \$mu_n=(1/n),sum_i=1^ndelta_X_i\$ the empirical measure. Then, \$mu_n(B)overseta.s.longrightarrowmu(B)\$ for all \$Bsubset S\$ and some (essentially unique) random probability measure \$mu\$. Denote by \$mathcalL(Z)\$ the probability distribution of any random variable \$Z\$. Under some assumptions on \$mathcalL(mu)\$, it is shown that eginequation* racanle hoigl[mathcalL(mu_n),,mathcalL(mu)igr]leracbnquad extandquad hoigl[mathcalL(mu_n),,mathcalL(a_n)igr]leraccn^u endequation* where \$ ho\$ is the bounded Lipschitz metric and \$a_n(cdot)=Pigl(X_n+1incdotmid X_1,ldots,X_nigr)\$ is the predictive measure. The constants \$a,,b,,c&gt;0\$ and \$uin (rac12, 1]\$ depend on \$mathcalL(mu)\$ and card\$,(S)\$ only.

### Rate of convergence of empirical measures for exchangeable sequences

#### Abstract

Let \$S\$ be a finite set, \$(X_n)\$ an exchangeable sequence of \$S\$-valued random variables, and \$mu_n=(1/n),sum_i=1^ndelta_X_i\$ the empirical measure. Then, \$mu_n(B)overseta.s.longrightarrowmu(B)\$ for all \$Bsubset S\$ and some (essentially unique) random probability measure \$mu\$. Denote by \$mathcalL(Z)\$ the probability distribution of any random variable \$Z\$. Under some assumptions on \$mathcalL(mu)\$, it is shown that eginequation* racanle hoigl[mathcalL(mu_n),,mathcalL(mu)igr]leracbnquad extandquad hoigl[mathcalL(mu_n),,mathcalL(a_n)igr]leraccn^u endequation* where \$ ho\$ is the bounded Lipschitz metric and \$a_n(cdot)=Pigl(X_n+1incdotmid X_1,ldots,X_nigr)\$ is the predictive measure. The constants \$a,,b,,c>0\$ and \$uin (rac12, 1]\$ depend on \$mathcalL(mu)\$ and card\$,(S)\$ only.
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Berti, Patrizia; Pratelli, Luca; RIGO, PIETRO
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11585/733368`
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