ON ALMOST AND ALMOST STATISTICAL CONVERGENCE IN CONE METRIC SPACES

Source abstract

We introduce the notions of almost convergence and almost statistical convergence in cone metric spaces, combining cone-ordered distances with shift-uniform Cesàro averaging and density methods. We establish basic structural properties including uniqueness of limits and order-dominance invariance, and show that almost convergence implies almost statistical convergence under a natural norm-control condition. We also provide examples showing that the converse fails in general. We further study almost statistical sequential compactness and obtain boundedness and closedness properties. In addition, we introduce almost statistically Cauchy sequences and a corresponding notion of completeness. We prove that in complete cone metric spaces, every almost statistically Cauchy sequence admits a convergent subsequence, and hence has an ordinary limit point. These results provide a unified framework for shift-uniform convergence in cone metric spaces and suggest directions for further research.