Some Properties of g-Covariance and g-Correlation Coefficient

A new nonlinear covariance based on -expectation, -covariance, is introduced and some properties of -covariance including commutativity, homogeneity, additivity are studied. According to these results, the sufficient and necessary condition of -covariance satisfied homogeneity and additivity is obtained. That is , a linear function of . Correlation coefficient based on -expectation does not rang from -1 to 1 and it does not reflect the linear relationship between two random variables, either.