Extension of Kosambi-Karhunen-Loeve expansion in L

This study presents an extension of the Kosambi-Karhunen-Loève (KKL) decomposition to accommodate stochastic processes with potentially infinite variance. We propose a nonlinear homeomorphic transformation upon which the classical KKL expansion may be applied. Subsequently, we establish the existence of specific subsets wherein this transformation preserves isometric properties. The theoretical framework is illustrated through an application to symmetric a-stable processes, demonstrating the practical utility of the proposed methodology.