Some multivariate inequalities with applications
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SOME MULTIVARIATE INEQUALITIES WITH APPLICATIONS Sana Louhichi (Paris) and Sofyen Louhichi (Paris) [Communicated by: J´ ozsef Fritz ]
Abstract Let Xn = (X1 , . . . , Xn ) be a random vector. Suppose that the random variables (Xi )1≤i≤n are stationary and fulfill a suitable dependence criterion. Let f be a real valued function defined on n having some regular properties. Let Yn be a random vector, independent of Xn , having independent and identically distributed components. We control | (f (Xn )) − (f (Yn ))|. Suitable choices of the function f yield, under minimal conditions, to rates of convergence in the central limit theorem, to some moment inequalities or to bounds useful for Poisson approximation. The proofs are derived from multivariate extensions of Taylor’s formula and of the Lindeberg decomposition. In the univariate case and in the mixing setting the method is due to Rio (1995).
1. Introduction
Let µn be a probability measure on n ; we suppose that µn is the law of a ˜n vector Xn = (X1 , . . . , Xn ) which have dependent components (Xi )1≤i≤n . Let µ denote a product probability measure on n ; more precisely µ ˜n is the law of a vector Yn = (Y1 , . . . , Yn ) which have independent and identically distributed (i.i.d.) components. A special case is when the random variables Xi and Yi have the same distributions for each 1 ≤ i ≤ n. The purpose of this paper is to evaluate the quantity
(f (Xn)) − (f (Yn )) ,
(1)
which is equal to µn (dy) = [f (x) − f (y)]µn (dx)˜ µn (dy), f (x)µn (dx) − f (y)˜ Mathematics subject classification number: 60F05. Key words and phrases: Lindeberg decomposition, multivariate inequalities, dependence, Central Limit Theorem, moment inequalities, Rosenthal inequalities, Poisson approximation. 0031-5303/2005/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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s. louhichi and s. louhichi
for functions f belonging to the set F (B2 , B3 ) of real valued functions defined on n , which are three times differentiable and fulfill 2 ∂ f ∂3f < ∞. < ∞, B3 = max (2) B2 = max 1≤i,j≤n ∂xi ∂xj 1≤i,j,k≤n ∂xi ∂xj ∂xk
When we are interested to the Central Limit Theorem, we will consider the following functions f x1 + . . . + xn √ f (x1 , . . . , xn ) = g , (3) n
for suitable real-valued function g belonging to the set G of functions defined on having bounded second and third derivatives. In this case 3/2 (3) 1 1 g . (4) B2 = g ∞ , B3 = ∞ n n √ Another normalization an rather than n can be used. We can also define f as f (x1 , . . . , xn ) = g(x1 + . . . + xn ), for g ∈ G. In this case B2 = g ∞ ,
(5)
B3 = g (3) ∞ .
Some interpolation techniques allow to include the following special choice of g : g(x) = xr , for some positive real number r. This choice of g leads to some moments inequalities (close to Rosenthal inequalities) for sums of the components of the random vector Xn . Another function with independent interest is f (x1 , . . . , xn ) =
n
g(xi ),
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