Summary
We use the theory of large deviations on function spaces to extend Erdös and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths {W N, n } defined by\(W_{N,n} (t,\omega ) = \frac{{S_{n + [h(N)t]} (\omega ) - S_n (\omega )}}{{h(N)}} + \frac{{h(N)t - [h(N)t]}}{{h(N)}}X_{n + [h(N)t] + 1} (\omega ),\) where\(S_n = \frac{1}{n}\sum\limits_1^n {X_i }\), {X i : i ≧1} is an iid sequence of random variables, andh(N)=[clogN] is relatively compact; the limit set is given by the set [x∶I *(x)≦1/c] whereI *(x) = ∫ 10 I(x′(t))dt andI is Cramér's rate function.
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Sanchis, G.R. A functional limit theorem for Erdös and Rényi's law of large numbers. Probab. Th. Rel. Fields 98, 1–5 (1994). https://doi.org/10.1007/BF01311345
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DOI: https://doi.org/10.1007/BF01311345
Mathematics Subject Classification (1991)
- P54C40
- 14E20
- S46E25
- 20C20