Karhunen–Loève Expansion Theory in AI–Blockchain Supply Chains, and Sustainable Logistics

Abstract

The Karhunen–Loève (KL) expansion provides an optimal orthogonal series representation of second-order stochastic processes and random fields. This paper presents a complete mathematical formulation of the KL theory using native Office Math Markup Language (OMML) equations, covering the governing Fredholm integral eigenvalue problem, the expansion and its coefficients, truncation error bounds, and the normalized representation. Building upon this theoretical foundation, we systematically explore practical applications of the KL expansion in contemporary supply-chain research themes: quantum-inspired optimization and uncertainty quantification in logistics networks; construction of resilient and adaptive digital twins for global supply chains; stochastic modelling supporting artificial-intelligence and blockchain integration for transparency, traceability and resilience; and uncertainty-aware modelling in sustainable and green logistics. The KL expansion emerges as a rigorous, computationally tractable tool for dimensionality reduction, random-field generation and risk quantification across these domains, thereby bridging classical stochastic process theory with the emerging requirements of Industry 4.0 and quantum-era logistics systems. Keywords— Karhunen–Loève expansion; stochastic processes; uncertainty quantification; digital twin; supply chain resilience; quantum logistics; green logistics; AI–blockchain integration.