关键词: 复g_λ随机变量, 准范数, 噪声, 经验风险最小化原则, 关键定理

Abstract: The key theorem plays an important role in the statistical learning theory.However，the researches about it at present mainly focus on real random variable and the samples which are supposed to be noise-free.In this paper，the definitions of complex g_λ variable and primary norm are introduced.Then，the definitions of the empirical risk functional，the expected risk functional and empirical risk minimization principle about g_λ samples corrupted by noise are proposed.Finally，the key theorem of learning theory about complex g_λ samples corrupted by noise is proposed and proved.The investigations help lay essential theoretical foundations for the systematic and comprehensive development of the statistical learning theory of complex g_λ samples.

Key words: complex g_λ variable, primary norm, noise, empirical risk minimization principle, the key theorem