Bounds for geometric sums used for evaluation of reliability of regenerative models
Let \(\{X_i\}\), \(\{Y_i\}\), \(Z\), and \(N\) be independent nonnegative random variables, where each of the sequences \(\{X_i\}\) and \(\{Y_i\}\) consists of identically distributed random variables, and \(N\) is a geometric random variable such that \({\mathbf P}(N=k)=(1-p)p^{k-1}\). Denote \[ W(x)={\mathbf P}\left(\sum^N_{i=1}X_i+\sum^{N-1}_{i=1}Y_i+Z\leq x\right)\quad\text{and}\quad W_0(x)={\mathbf P}\left(\sum^N_{i=1}X_i\leq x\right). \] The authors obtain bounds on \(\delta(x)=W_0(x)-W(x)\). For example, they show, among other things, that if \(W_0\) has a density \(w_0\), then \[ \delta(x)\leq\overline w_0(x)\Bigl({\mathbf E}_Z(x)+{p\over 1-p}{\mathbf E}_Y(x)W_0(x)\Bigr), \] where \({\mathbf E}_Y(x)=\int^x_0{\mathbf P}(Y_1>u)du\), \({\mathbf E}_Z(x)=\int^x_0{\mathbf P}(Z>u)du\), and \(\overline w_0(x)=\sup_{u\leq x}w_0(u)\). They also give upper bounds on \(\overline w_0(x)\) and on \(W_0(x)\) in order to estimate the upper bound above. These bounds are the following. If \(X_1\) has distribution function \(F\) and density function \(f\), then \(\overline w_0(x)\leq{(1-p)\overline f(x)\over 1-pF(x)}\), where \(\overline f(x)=\sup_{u\leq x}f(u)\). If there exists a positive \(\varepsilon\) such that \(p{\mathbf E}e^{\varepsilon X_1}=1\), and if \(X_1\) is NBU (new better than used), then \(W_0(x)\leq 1-e^{-\varepsilon x}\). The authors compare some of their bounds with a bound of \textit{M. Brown} [Ann. Probab. 18, No. 3, 1388-1402 (1990; Zbl 0709.60016)].
scientific article; zbMATH DE number 1295564