Weighted approximation of the renewal spacing processes (Q2366543)

Let \(\omega\), \(\omega_ 1\), \(\omega_ 2,\dots\) be i.i.d. r.v.'s with mean \(E \omega=1\), variance \(0<\text{Var} \omega<\infty\) and common distribution function \(F\), which is twice differentiable except maybe at a finite number of points. Put \(S_ n=\omega_ 1 +\cdots+\omega_ n\), \(D_{i,n}=\omega_ i/S_ n\), \(Q(s)=\inf\{x:F(x)\geq s\}\), \(D_ n(s)=n^{-1}\sum^ n_{i=1} [nD_{i,n} \leq Q(s)]\) and \(D_ n^ \leftarrow (t)=\inf\{s:D_ n(s)\geq t\}\). Limiting properties of the processes \(a_ n(s)=n^{1/2} (D_ n(s)-s)\) and \(b_ n(s)=n^{1/2} (D_ n^ \leftarrow (s)-s)\), formed by means of the empirical distribution functions \(D_ n\) and the corresponding quantile processes \(D_ n^ \leftarrow\), are investigated. The main results state that, if for some \(\nu \in[0,1/4)\), \(\int[F(x)(1-F(x))]^{1/2-\nu}<\infty\), then under suitable assumptions on \(F\) one can construct a sequence of Brownian bridges \(\{B_ n,n \geq 1\}\), such that \[ n^ \nu \sup_{1/n \leq s \leq(n-1)/n} \left| {a_ n(s)-\Gamma_{B_ n}(s) \over [s(1- s)]^{1/2-\nu}} \right|=O_ P(1), \tag{i} \] and under some additional assumptions, \[ n^ \nu \sup_{1/n \leq s \leq(n-1)/n} \left| {b_ n(s)+\Gamma_{B_ n}(s) \over[s(1-s)]^{1/2-\nu}} \right|=O_ P(1), \tag{ii} \] where \(\Gamma_ B(s)=B(s)-Q(s) F'(Q(s))\int^ 1_ 0B(t)dQ(t)\). Moreover, denote by \(D_{(1),n} \leq \cdots \leq D_{(n),n}\) the order statistics of the renewal spacings \(D_{i,n}\). Then under the assumptions of (i) and (ii) resp., \[ \begin{aligned} \sup_{0<s \leq 1} D_ n(s)/s+\sup_{0 \leq s<1} (1-D_ n(s))/(1- s)=O_ P(1),\\ \sup_{F(nD_{(1),n}) \leq s \leq 1} s/D_ n(s)+\sup_{0 \leq s \leq F(nD_{(n),n})} (1-s)/(1-D_ n(s))=O_ P(1),\end{aligned}\tag{i\('\)} \] and for any \(\lambda>0\), \[ \begin{aligned} \sup_{\lambda/n \leq s \leq 1}D_ n^ \leftarrow (s)/s+ \sup_{0 \leq s \leq 1-\lambda/n} (1-D_ n^ \leftarrow (s))/(1-s)=O_ P(1),\\ \sup_{0<s<1} s/D_ n^ \leftarrow (s)+\sup_{0<s<1} (1-s)/(1-D_ n^ \leftarrow (s))=O_ P(1).\end{aligned}\tag{ii\(''\)} \] .