A note on tightness
Let \(T=\{(s,t):0\leq s\leq t\leq 1\}\), and let \(\mathcal T\) be the set of functions from \(T\) to \([0,\infty)\) that are decreasing in the first argument and increasing in the second one. Two concepts: ``a function \(\varphi\in{\mathcal T}\) is said to be subservient to another function \(\rho\in{\mathcal T}\)'', and ``a function \(\rho\in{\mathcal T}\) is said to be diagonally vanishing'' are introduced (the details are in the paper). In terms of those concepts the Billingsley's classical tightness criterion for sequences of càdlàg processes on \([0,1]\) is extended to the following: \[ P[|X_n(u)-X_n(s)|\geq\lambda, |X_n(t)-X_n(u)|\geq\lambda]\leq\lambda^{-2\gamma}\varphi_n(s,t), \] \(\lambda>0\), \(n\geq 1\), \(0\leq s\leq u\leq t\leq 1\), \(t-s<\Delta\), for some \(\Delta>0\), \(\gamma>0\) and a sequence \((\varphi_n)\) of elements of \(\mathcal T\) such that \(\varphi_n\) is \((1/(2\gamma+1),\Delta)\)-subservient to \(\rho_n\) for all \(n\) superior to some \(n_0\), and that the sequence \((\rho_n)\) converges pointwise to a limiting diagonally vanishing function \(\rho\in{\mathcal T}\).
scientific article; zbMATH DE number 896097