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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 $$M(t)$$, $$t>0$$.

##### MSC:
 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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