A functional extremal criterion
Let \({\mathcal N}=\{(t_k, X_k):\, k\geq 1\}\) be a point process with time space \([0, \infty)\) and state space \([0, \infty)^d\), where \(\{t_k\}\) are distinct nonrandom time points monotonically increasing to \(\infty\). \(\{X_k\}\) are independent and identically distributed random vectors on a given probability space with values in \([0,\infty)^d\) and with common distribution function nondefective at \(+\infty\). Assume that almost all realizations of \({\mathcal N}\) are Radon measures on \({\mathcal S}=[0,\infty)\times E\), where \(E=[0,\infty]^d\setminus\{0\}\). The authors consider two random processes associated with \({\mathcal N}\), namely, the extremal process \(X(t) =\{\vee X_k:\, t_k\leq t\}\) and the process \(Z(t)= \{\sum X_k:\, t_k\leq t\}\) with additive increments. Denote by \({\mathcal M}\) the set of all nondecreasing, right-continuous functions \(y: (0,\infty) \to [0, \infty)^d\). Then the set \({\mathcal P}\) of all probability measures on \({\mathcal M}\) is compact. Let \(\{P_n\}\) be a sequence of probability measures on \({\mathcal M}\). \(\{P_n\}\) is weakly convergent to \(P\in {\mathcal P}\), briefly \(P_n \Rightarrow P\), if \(\int \varphi dP_n \to \int \varphi dP\) for bounded \(\varphi: {\mathcal M} \to R\) which are continuous in the weak topology of \({\mathcal M}\). Denote by \({\mathcal P}_e\) and \({\mathcal P}_s\) the subsets of \({\mathcal P}\) corresponding to an extremal process (with independent max-increments) and to a sum process (with independent additive increments), respectively. \textit{A. A. Balkema} and \textit{E. I. Pancheva} [Commun. Stat., Theory Methods 25, No. 4, 737--758 (1996; Zbl 0875.60010), Theorem 6.4] have shown that the space \({\mathcal P}_e\) with the topology of weak convergence is closed in \({\mathcal P}\). The same is also true for \({\mathcal P}_s\). So, the weak convergence of extremal processes \(Y_n\Rightarrow Y\) and of sum processes \(S_n \Rightarrow S\) is equivalent to the convergences in distribution. Let \(\zeta_n(t, x) =(\tau_n(t), u_n(x))\) be an increasing in \(n\) sequence of mappings, continuous and strictly increasing in each coordinate. Suppose \(\{\zeta_n\}\) is regular in the sense that there exists a pointwise limit of \(\zeta_n^{-1}\circ \zeta_{[ns]}\) for \(n\to \infty\) and \(s>0\) which is again continuous and strictly increasing [see \textit{E. I. Pancheva}, J. Math. Sci., New York 92, No. 3, 3911--3920 (1998; Zbl 0926.60045)]. Let \(Y_n(t) =\{\vee u_n^{-1}(X_k): t_k\leq \tau_n(t)\} \Rightarrow Y(t)\), \(n\to \infty\), to a nondegenerate extremal process \(Y\) with initial value \(Y(0) =0\) a.s. and let \(S_n(t) =\{\sum u_n^{-1}(X_k): t_k\leq \tau_n(t)\}\) be the associated processes with additive increments. Not that the space changes \(\{u_n\}\) preserve the max-operator, i.e. \(u_n^{-1}(\vee X_k) = \vee u_n^{-1}(X_k)\), but do not preserve (in general) the summing operator. Hence, \(Y_n(t) = u_n^{_1}\circ X\circ \tau_n(t)\), but \(S_n(t)\neq u_n^{-1}\circ Z\circ \tau_n(t)\) in general. If \(u_n\) preserves both operations \(\vee\) and \(\sum\), then \(u_n\) is a scale change and the convergence \(S_n = u_n^{_1}\circ Z\circ \tau_n\Rightarrow S\) implies that \(S\) is a self-similar process [see \textit{J. Lamperti}, Trans. Am. Math. Soc. 104, 62--78 (1962; Zbl 0286.60017)]. The main result of the paper, proved in Section 2, concerns the convergence \(S_n\Rightarrow S\), if given \(Y_n\Rightarrow Y\). The authors call it a functional extremal criterion (for the convergence \(S_n\Rightarrow S\)), having in mind the extremal criterion of \textit{M. Loève} [``Probability theory'' (1960; Zbl 0095.12201)].
scientific article; zbMATH DE number 2183273