Probability Theory and Related Fields

, Volume 76, Issue 4, pp 429-438

First online:

Elliptically contoured distributions

  • Yehoram GordonAffiliated withDepartment of Mathematics, Technion, Israel Institute of Technology

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Given two covariance matricesR andS for a given elliptically contoured distribution, we show how simple inequalities between the matrix elements imply thatE R(f)≦E S(f), e.g., whenx=(xi1,i2,...,in) is a multiindex vector and
$$f(x) = \mathop {\min }\limits_{i_1 } \mathop {\max }\limits_{i_2 } \mathop {\min }\limits_{i_3 } \max ...x_{i_{1,...,} i_n } ,$$
orf(x) is the indicator function of sets such as
$$\mathop \cap \limits_{i_1 } \mathop \cup \limits_{i_2 } \mathop \cap \limits_{i_3 } \cup ...[x_{i_{1,...,} i_n } \mathop< \limits_ = \lambda _{i_{1,...,} i_n } ]$$
of which the well known Slepian's inequality (n=1) is a special case.