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A functional limit theorem for Erdös and Rényi's law of large numbers
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  • Published: March 1994

A functional limit theorem for Erdös and Rényi's law of large numbers

  • Gabriela R. Sanchis1 

Probability Theory and Related Fields volume 98, pages 1–5 (1994)Cite this article

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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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Authors and Affiliations

  1. Department of Mathematical Sciences, Elizabethtown College, 17022, Elizabethtown, PA, USA

    Gabriela R. Sanchis

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  1. Gabriela R. Sanchis
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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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  • Received: 10 August 1992

  • Revised: 07 June 1993

  • Issue Date: March 1994

  • DOI: https://doi.org/10.1007/BF01311345

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Mathematics Subject Classification (1991)

  • P54C40
  • 14E20
  • S46E25
  • 20C20
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