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Logarithmic law: twenty years later. (English. Ukrainian original) Zbl 0955.60054
Theory Probab. Math. Stat. 60, 19-27 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 17-25 (1999).
Let \(\Xi=\bigl(\xi_{ij}^{(n)}\bigr)_{i,j=1}^{n}\) be a random matrix such that the random variables \(\xi_{ij}^{(n)}\) are independent, \(E\xi_{ij}^{(n)}=0\), \(\operatorname {Var}\xi_{ij}^{(n)}=1.\) Approximately twenty years passed since the author has published the logarithmic law for random determinants. He proved that under the assumption \(E\bigl[\xi_{ij}^{(n)}\bigr]^4=3\), \(i,j=1,2,\ldots\), the following assertion holds true \[ \lim_{n\to\infty}P\Biggl({\ln \det\Xi^2 - \ln[(n-1)!]\over \sqrt{2\ln n}}<x\Biggr)= {1\over \sqrt{2\pi}}\int_{-\infty}^x \exp\Biggl( {u^2\over 2}\Biggr) du. \] The assertion \(E[\xi_{ij}^{(n)}]^4=3\) is seemed very strict. In this paper the author proves the logarithmic law without this assertion. He proves that the logarithmic law holds true under the condition \(\sup_n \sup_{i,j} E|\xi_{ij}^{(n)}|^{4+\delta}<\infty\) for some \(\delta>0\).
60G60 Random fields
62M20 Inference from stochastic processes and prediction