On functional versions of the arc-sine law
Suppose \(R(t)\) and \(R_n(t)\), \(n\geq 1\), are random processes in \(D(0,1)\) such that \(R_n(t)\) converges weakly to \(R(t)\) in \(D(0,1)\). For \(z\in(-\infty,\infty)\) define \(\Lambda_n(z)= \int^1_0\mathbf{1}\{R_n(t)\leq z\}\,dt\) and similarly \(\Lambda(z)\) for the process \(R(t)\). Under some additional conditions on path properties of the limit proces \(R(t)\) it is proved that \(\Lambda_n(z)\) converges weakly to \(\Lambda(z)\) in \(D[-\infty,\infty]\). This general theorem contains the following important example. Let \(X_1, X_2,\dots\) are i.i.d. random variables with mean 0 and the unit variance. Then \[ {1\over n}\sum^n_{k=1}\mathbf{1}\{(X_1,+\cdots+ X_k)/\sqrt{k}\leq z\} \] converges weakly in \(D[-\infty,\infty]\) to \(\int^1_0 \mathbf{1}\{W(t)/\sqrt{t}\leq z\}\,dt\), where \(W(t)\) is a standard Brownian motion. Taking \(z= 0\) one obtains the classical Lévy's first arc-sine law. The general theorem is used to derive several convergence results for normalized partial sums of dependent random variables both for Gaussian and normalized Lévy limit processes \(R(t)\) as well as for extreme-value statistics of i.i.d. random variables.
scientific article; zbMATH DE number 5700610
scientific article