For each s : 1,2... let n: r?(s) be a posit ive integer such that n/s --+ y > 0 as s --+ cc. Let V" be an n X s matrix whose entries are i.i.d. N(0, 1) random variables and let M": (I/s)V"V:.The random matrix V,V: is commonly referred to as the Wishart matrix W(1,, s). It is well known [Mar0enko and Pastur (1967), Wachter (1978)] that the empirical distribution function 4 of the eigenvalues of M, t.F"(r) = (I/n) x (number of eigenvalues of M " < *)l converges almost r.rr"ly ur r -+ oo to a nonrandom probability distribution function { having a density with positive support on [(1 tF)',(l + ,F>t], and when yi r, { yields additional mass on {0}. It is also known fGeman (1980)] that the maximum eigenvalue I(** of M" converges a.s. to (1 + ,F)'as s -+ oo. [The statement of thi. r"rult in Geman (1980) has all the M, constructed from one doubly infinite array of i.i.d. random variables. However, it is obvious from the proof that no relation on the entries of V" for different s is needed.] These results are established under assumptions more general on the entries of % than Gaussian distributed, involving conditions on the moments of these random variables. The present paper will prove the following

Jack W. Silverstein

The Annals of Probability

概率年鉴发表现代概率论的研究论文，它与其他数学领域的关系，以及它在物理和生物科学中的应用。重点是重要性、兴趣和独创性——形式上的新颖性和正确性不足以发表。 《年鉴》还将发表权威的评论论文和对蓬勃发展领域的调查。

The Annals of Probability publishes research papers on modern probability theory, exploring its connections with other areas of mathematics and its applications in physical and biological sciences. The journal emphasizes the importance, interest, and originality of the papers; mere novelty and correctness are insufficient for publication. Additionally, the Annals will publish authoritative review papers and surveys of thriving fields.

《Annals of Probability》发表现代概率论的研究论文，探讨其与其他数学领域的关系，以及在物理和生物科学中的应用。期刊重点关注论文的重要性、趣味性和独创性，单纯的形式新颖性和正确性不足以满足发表要求。此外，《Annals of Probability》还会刊登权威的评论论文和对快速发展的领域的综述。

Category	Quartile
数学	1
数学, 统计学与概率论	1

The Smallest Eigenvalue of a Large Dimensional Wishart Matrix

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Category	Quartile
STATISTICS & PROBABILITY	1