Summary
Let X 1 , X 2 , ..., X n be i.i.d. random vectors in R p where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p 2/n tends to zero. Furthermore, an example is presented with X i having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean \((\sqrt {\text{n}} \overline {\text{X}} )\) can not hold if p 2/n does not tend to zero.
References
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Research supported in part by National Science Foundation Grants MCS 80-02340, MCS 83-01834, and DMS 85-03785
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Portnoy, S. On the central limit theorem in R p when p→∞. Probab. Th. Rel. Fields 73, 571–583 (1986). https://doi.org/10.1007/BF00324853
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DOI: https://doi.org/10.1007/BF00324853
Keywords
- Normal Distribution
- Generate Function
- Stochastic Process
- Probability Theory
- Limit Theorem