A generalization of the global limit theorems of R. P. Agnew

For distribution functions { F n , n ≥ 0 } , the relationship between the weak convergence of F n to F 0 and the convergence of ∫ R ϕ ( | F n − F 0 | ) d x to 0 is studied where ϕ is a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby generalizing the so-called global limit theorems of Agnew wherein ϕ ( t ) = | t | r . The sufficiency results are shown to be sharp and, as a special case, yield a global version of the central limit theorem for independent random variables obeying the Liapounov condition. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to Lebesgue measure on the line.