In the paper some multivariate power generalizations of Chebyshev’s inequality and their improvements will be presented with extension to a random vector with singular covariance matrix. Moreover, for these generalizations, the cases of the multivariate normal and the multivariate t distributions will be considered. Additionally, some financial application will be presented.

multivariate Chebyshev’s inequality, Mahalanobis distance, multivariate normal distribution, multivariate t distribution

BUDNY, K., (2014). A generalization of Chebyshev's inequality for Hilbert-spacevalued random elements. Statistics and Probability Letters, 88, pp. 62–65.

BUDNY, K., (2016). An extension of the multivariate Chebyshev’s inequality to a random vector with a singular covariance matrix, Communications in Statistics – Theory and Methods, 45 (17), pp. 5220–5223.

CHEN, X., (2007). A new generalization of Chebyshev inequality for random vectors. Available at: https://arxiv.org/abs/0707.0805 [Accessed 5 July 2007].

CHEN, X., (2011). A new generalization of Chebyshev inequality for random vectors. Available at: https://arxiv.org/abs/0707.0805v2 [Accessed 24 June 2011].

JOHNSON, N.L., KOTZ, S., BALAKRISHNAN, N., (1994). Continuous univariate distribution. Vol. 1, 2nd ed. John Wiley & Sons Inc.

JOHNSON, N.L., KOTZ, S., BALAKRISHNAN, N., (1995). Continuous univariate distribution, Vol. 2, 2nd ed. John Wiley & Sons Inc.

KRITZMAN, M., Li, Y., (2010). Skulls, financial turbulence, and risk management. Financial Analysts Journal, 66 (5), pp. 30–41.

KOTZ, S., BALAKRISHNAN, N., JOHNSON, N.L., (2000). Continuous multivariate distribution, Vol. 1: Models and applications, 2nd ed. John Wiley & Sons Inc.

LIN, P., (1972). Some characterizations of the multivariate t distribution, Journal of Multivariate Analysis, 2, pp. 339–344.

LOPERFIDO, N., (2014). A probability inequality related to Mardia’s kurtosis. In: C. Perna, M. Sibillo (eds.). Mathematical and statistical methods of actuarial science and finance, Springer: Springer International Publishing Switzerland 201. pp. 129–132.

MARDIA, K.V., (1970). Measures of multivariate skewness and kurtosis with applications, Biometrika, 57 (3), pp. 519–530.

MARSHALL, A., OLKIN, I., (1960). Multivariate Chebyshev inequalities, The Annals of Mathematical Statistics, 31, pp. 1001–1014.

NAVARRO, J., (2014). Can the bounds in the multivariate Chebyshev inequality be attained? Statistics and Probability Letters, 91, pp. 1–5.

NAVARRO, J., (2016). Avery simple proof of the multivariate Chebyshev’s inequality. Communications in Statistics – Theory and Methods, 45 (12), pp. 3458–3463.

OLKIN, I., PRATT, J.W., (1958). A multivariate Tchebycheff inequality. The Annals of Mathematical Statistics, 29, pp. 226–234.

OSIEWALSKI, J., TATAR, J., (1999). Multivariate Chebyshev inequality based on a new definition of moments of a random vector, Przegląd Statystyczny (Stat. Rev.), 2, pp. 257–260.

PEARSON, K., (1919). On generalised Tchebycheff theorems in the mathematical theory of statistics, Biometrika,12(3–4), pp. 284–296.

STÖCKL, S., HANKE, M., (2014). Financial applications of the Mahalanobis distance, Applied Economics and Finance, 1 (2), pp. 78–84.

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