Limit theorems for maxima of independent random sums (Q2711142)

Let \(X_{ij}\), \(i,j\geq 1\), be i.i.d. random variables with zero expectation and unit variance. Put \(Z_{mn} \sqrt{n}= \max \{X_{i1}+ \cdots+ X_{in}: i=1,\dots, m\}\). The paper derives conditions such that NEWLINE\[NEWLINEP(a_m (Z_{mn}- b_m)\leq x)\to \exp \{-e^{-x}\}\tag \(*\) NEWLINE\]NEWLINE as \((m,n)\to \infty\), where \(a_m\) and \(b_m\) are the norming and centering constants when the \(X_{ij}\) are standard normal. Conditions on moments and on the order of \(m\to\infty\) as \(n\to\infty\) are sufficient. Under weaker conditions \((*)\) holds with \(a_m\), \(b_m\) replaced by \(\alpha_{mn}\sim a_m\) and \(\beta_{mn}\). Under virtually no condition there still is some limiting behaviour. Application: norms of random matrices.