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A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

  • Patrizia Berti (a1), Irene Crimaldi (a2), Luca Pratelli (a3) and Pietro Rigo (a4)

Abstract

Let X n be a sequence of integrable real random variables, adapted to a filtration (G n ). Define C n = √{(1 / n)∑ k=1 n X k − E(X n+1 | G n )} and D n = √n{E(X n+1 | G n ) − Z}, where Z is the almost-sure limit of E(X n+1 | G n ) (assumed to exist). Conditions for (C n , D n ) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑ k=1 n X_k - Z} = C n + D n N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

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Copyright

Corresponding author

Postal address: Dipartimento di Matematica Pura ed Applicata ‘G. Vitali’, Università di Modena e Reggio-Emilia, via Campi 213/B, 41100 Modena, Italy. Email address: patrizia.berti@unimore.it
∗∗ Postal address: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy. Email address: crimaldi@dm.unibo.it
∗∗∗ Postal address: Accademia Navale, viale Italia 72, 57100 Livorno, Italy. Email address: pratel@mail.dm.unipi.it
∗∗∗∗ Postal address: Dipartimento di Economia Politica e Metodi Quantitativi, Università di Pavia, via S. Felice 5, 27100 Pavia, Italy. Email address: prigo@eco.unipv.it

References

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Keywords

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A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

  • Patrizia Berti (a1), Irene Crimaldi (a2), Luca Pratelli (a3) and Pietro Rigo (a4)

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