## Abstract

Let X^{(n)} = (X_{ij}) be a p × n data matrix, where the n columns form a random sample of size n from a certain p-dimensional distribution. Let R^{(n)} = (ρ_{ij}) be the p × p sample correlation coefficient matrix of X^{(n)}, and S^{(n)} = (1/n)X^{(n)}(X^{(n)})*-X̄X̄* be the sample covariance matrix of X^{(n)}, where X̄ is the mean vector of the n observations. Assuming that X_{ij} are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R^{(n)} converges almost surely to the limit (1-√c)^{2} as n → ∞ and p/n → c ∈ (0,∞). We accomplish this by showing that the smallest eigenvalue of S^{(n)} converges almost surely to (1-√c)^{2}.

Original language | English (US) |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Theoretical Probability |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2010 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty

## Keywords

- Random matrix
- Sample correlation coefficient matrix
- Sample covariance matrix
- Smallest eigenvalue