Asymptotic behavior of large deviation probabilities for two weighted sums of random variables (Q6632311)

Let \(\{X_i\}_{1\leq i\leq n_1}\) are i.i.d. nonlattice variables and \(\{Y_j\}_{1\leq j\leq n_2}\) are i.i.d. nonlattice variables such that \(\mathbf{E}(X^2_1e^{h^{-}_1X_1}), \mathbf{E}(X^2_1e^{h^{+}_1X_1}), \mathbf{E}(Y^2_1e^{h^{-}_2Y_1})\) and \({\mathbf{E}(Y^2_1e^{h^{+}_2Y_1})}\) are finite for some constants \({h^{-}_k<0<h^{+}_k}, {k=1,2}.\) Let \N\[\NS_{n_1,1}=\sum_{i=1}^{n_1}a_{i,n_1}X_i,\ \ S_{n_2,2}=\sum_{j=1}^{n_2}b_{j,n_2}Y_j ,\N\]\Nwhere \({a_{i,n_1}=f(i/n_1)}\) and \({b_{j,n_2}=g(j/n_2)}\) for some positive twice continuously differentiable functions \(f,g\) on \([0,1].\) Under some conditions the author obtains asymptotics for \(\mathbf{P}(S_{n_1,1}>S_{n_2,2})\) as \({n_1,n_2\to \infty}\) and \({\frac{n_1}{n_1+n_2}\to p\in(0,1)}.\) The result is applied to the gladiator game model introduced in [\textit{K. S. Kaminsky} et al., Aust. J. Stat. 26, 111--118 (1984; Zbl 0558.62016)].