The limit of the smallest singular value of random matrices with i.i.d. entries

Konstantin Tikhomirov

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

Let {aij} (1≤i,j<∞) be i.i.d. real-valued random variables with zero mean and unit variance and let an integer sequence (Nm)m=1 satisfy m/Nm→z for some z∈(0,1). For each m∈N denote by Am the Nm×m random matrix (aij) (1≤i ≤ Nm, 1 ≤ j ≤ m) and let sm(Am) be its smallest singular value. We prove that the sequence (Nm-1/2sm(Am))m=1 converges to 1-√z almost surely. Our result does not require boundedness of any moments of aij's higher than the 2-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalAdvances in Mathematics
Volume284
DOIs
StatePublished - Oct 2 2015

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Random matrices
  • Smallest singular value

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