Affiliation:
1. Dipartimento di Matematica Pura ed Applicata “G. Vitali” Universita’ di Modena e Reggio-Emilia via Campi 213/B 41100 Modena Italy
2. Accademia Navale viale Italia 72 57100 Livorno Italy
3. Dipartimento di Matematica “F. Casorati” Universita’ di Pavia via Ferrata 1 27100 Pavia Italy
Abstract
Abstract
Let S be a finite set, (Xn
) an exchangeable sequence of S-valued random variables, and μn
= (1/n)
∑
i
=
1
n
$\sum\limits_{i=1}^{n} $
δ
Xi
the empirical measure. Then, μn
(B)
⟶
a
.
s
.
$\overset{a.s.}{\longrightarrow} $
μ(B) for all B ⊂ S and some (essentially unique) random probability measure μ. Denote by 𝓛(Z) the probability distribution of any random variable Z. Under some assumptions on 𝓛(μ), it is shown that
a
n
≤
ρ
[
L
(
μ
n
)
,
L
(
μ
)
]
≤
b
n
and
ρ
[
L
(
μ
n
)
,
L
(
a
n
)
]
≤
c
n
u
$$\begin{array}{}
\displaystyle \frac{a}{n}\le\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(\mu)\bigr]\le\frac{b}{n}\qquad\text{and}\qquad\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(a_n)\bigr]\le\frac{c}{n^u}
\end{array} $$
where ρ is the bounded Lipschitz metric and an
(⋅) = P(X
n+1 ∈ ⋅ | X
1, …, Xn
) is the predictive measure. The constants a, b, c > 0 and u ∈ (
1
2
$\frac{1}{2} $
, 1] depend on 𝓛(μ) and card (S) only.
Reference10 articles.
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3. Berti, P.—Pratelli, L.—Rigo, P.: Limit theorems for empirical processes based on dependent data, Electron. J. Probab. 17 (2012), 1–18.
4. Berti, P.—Pratelli, L.—Rigo, P.: Exchangeable sequences driven by an absolutely continuous random measure, Ann. Probab. 41 (2013), 2090–2102.10.1214/12-AOP786
5. Berti, P.—Pratelli, L.—Rigo, P.: Asymptotic predictive inference with exchangeable data, submitted (2016), currently available at http://www-dimat.unipv.it/rigo/smps2016.pdf
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