On Some Nonlinear Characteristics of the Center of Grouping of Random Variables
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Some Nonlinear Characteristics of the Center of Grouping of Random Variables V. L. Khatskevich1* 1 Military educational scientific center Air force “Air force academy named after N.E. Zhukovsky and Yu.A. Gagarin”, 54 a old Bolsheviks str., Voronezh, 394064 Russia
Received September 10, 2019; revised September 10, 2019; accepted December 18, 2019
Abstract—The paper devoted to different aspects of the theory of nonlinear mean random variables, i.e., the geometric mean, harmonic mean, and average power. DOI: 10.3103/S1066369X2008006X Key words: nonlinear mean of random variables, algebraic and extremal properties, inequalities.
INTRODUCTION The most well-known characteristics of the grouping center of random variables are their mean and median, but sometimes other characteristics are also useful. Some properties of the geometric mean, harmonic mean, and power-law mean of random variables are investigated below. The considered random variables are assumed to be positive and have finite mathematical expectations. The present paper introduces a general definition of a nonlinear mean of a random variable, which includes the concept of geometric mean, harmonic mean, and power mean. The problem of pointwise estimation of nonlinear means is considered. Extremal properties of nonlinear means are demonstrated. Algebraic properties similar to the mathematical expectation properties are given. Inequalities between the means are considered. Some of the properties given below are known for discrete analogs; others are not typical for them or contain differences. Comparison with known results is given along the way. Let (Ω, A, P ) be some probabilistic space (see, for example, [1], Ch. II, § 1). Here Ω is the space of elementary events, A is the σ-algebra of subsets from Ω, P is a probabilistic measure. Let R be the set of real numbers. We consider a random variable X : Ω → R. We denote by EX its mathematical expectation. We consider the Borelian function f (x) defined on the set of real numbers R. The function Y = f (X) of the random variable X is the random variable Y obtained by the superposition of the real function X = X(ω) given on the set Ω and the function f (x). That is, Y = f (X(ω)) = Y (ω). In this case, the mathematical expectation of Ef (X) is determined by ∞ f (x)dF (x), where F (x) is the distribution function of the Stiltjes–Lebesgue integral Ef (X) = −∞
the random variable X. Below we will use the inequalities between the random variables X and Y given on the same probabilistic space (Ω, A, P ), understanding X ≥ Y as X(ω) ≥ Y (ω) for all elementary events ω ∈ Ω. The geometric mean G(X) of the random variable X (X > 0) is given by the formula (see, for example, [2], Ch. 5, item 5.6.2) G(X) = eE(ln X) ,
(1)
the harmonic mean H(X) (X > 0) does by the formula H(X) = 1/(E(1/X)). *
E-mail: [email protected]
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(2)
ON SOME NONLINEAR CHARACTERISTICS
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Note that the geometric mean is used in the study of processes, the growth of which is proportional to the level already achieved (growth in population, gross product
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