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Extreme value asymptotics for multivariate renewal processes. (English) Zbl 0861.60065
Let \(\{X_n\}\) denote an i.i.d. sequence of \(d\)-dimensional random vectors with mean vector \(\mu\) and the dispersion matrix \(\Sigma\). Let \(S_n=X_1+X_2+\cdots+X_n\), \(n\geq 1\). Denote \(S_n'=(S_{n1},S_{n2},\dots,S_{nd})\) and \(\mu'=(\mu_1,\mu_2,\dots,\mu_d)\). Define the renewals \(N_i(t)=\min\{n\geq 1, S_{ni}\geq t\}\), \(M_i(t)=N_i(t\mu_i)-t\), \(i=1,2,\dots,d\), and \(M'(t)= (M_1(t),M_2(t),\dots,M_d(t))\). Some weak convergence results have been obtained for \(M(t)\) and its increments. An invariance principle has been established for \(M(t)\) in terms of a \(d\)-dimensional Wiener process. Further, similar weak convergence results have been established for \(\widehat M(t)\), an estimated version of \(M(t)\), \(t>0\).

60G70 Extreme value theory; extremal stochastic processes
60F05 Central limit and other weak theorems
60K05 Renewal theory
60F17 Functional limit theorems; invariance principles
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