Acta Mathematica Academiae Scientiarum Hungaricae 63. (1994)

1994 / 2. szám - Accardi, L. - Lu, Y. G.: Quantum central limit theorems for weakly dependent maps I

where / G L°°(S, O) and Acta Math. Hungar. 63 (2) (1994), 183-212. QUANTUM CENTRAL LIMIT THEOREMS FOR WEAKLY DEPENDENT MAPS I L. ACCARDI and Y.G. LU* (Roma) §0. Introduction Recall [3] that a stochastic process over a *-algebra B, indexed by a set T, is a triple (0.1) {A y>,(j,)teT} where A is a *-algebra (unless otherwise specified, all algebras in the present paper are complex, associative, with identity); ip a state on A; and jt : В —*■ —► A a ^-homomorphism. Every classical stochastic process (Xt) (t G T), from a probability space (Q,iF,P) to a state space (S,0) (a measurable space) naturally defines a structure as described above by choosing A = L°°(il,P,P) ; В = L°°(S,B), jf- f £ L°°(S, В) - jtU) := f о Xte T, P) and <p to be the integral with respect to the P-measure. Conversely, every triple of the form (0.1) with A and В abelian, determines a (unique up to isomorphism) classical stochastic process. Now let Г be a subset of the natural integers. The classical law of large numbers (resp. central limit theorem) studies the asymptotic behaviour (for ./V —> oo) of the normalized sums 7Ш = / f{x3) dp. Ju N / N *£/No> resp. £[/No) - /No)] 3=1 \ ^ ’ i=l * On leave of absence from Beijing Normítl University.

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