On Tridiagonal Correlation Matrices

We investigate the conditions under which symmetric tridiagonal matrices represent valid correlation matrices. By exploiting a recursive determinant relationship, we derive explicit sufficient conditions for positive definiteness and highlight connections with several existing criteria. For dimensions up to four, we precisely characterize feasible parameter regions, providing both analytical expressions and intuitive geometric interpretations. We pay special attention to two structured cases: stationary processes, where we establish a sharp necessary and sufficient bound, and alternating period-2 correlation structures, whose spectral properties yield exact semidefiniteness criteria. The derived results furnish practical guidelines for verifying the validity of banded correlation models in various applied contexts.