Uniform integrability for strong ratio limit theorems. III

Unlike Parts I and II of this paper [Theory Probab. Appl. 50, No. 3, 436–447 (2006); translation from Teor. Veroyatn. Primen. 50, No. 3, 517–532 (2005; Zbl 1120.60028) and Theory Probab. Appl. 55, No. 3, 473–484 (2011); erratum ibid. 56, No. 1, 178 (2012); translation from Teor. Veroyatn. Primen. 55, No. 3, 446–461 (2010; Zbl 1247.60040)], which dealt with strong limit theorems for ratios (SLTR) in the traditional sense, now we propose new SLTR with particular parametric sets. The peculiarity of these theorems is that their statements ignore some values of time parameter that form a particular set of density 0. SLTR of this type for random walks on unimodular locally compact groups are given as an illustration of the general theory.