**T. Shervashidze**

## On Bounds for the Characteristic Functions of
Some Degenerate Multidimensional Distributions

**abstract:**

We discuss an application of an inequality for the modulus of the characteristic
function of a system of monomials in random variables to the convergence of the
density of the corresponding system of the sample mixed moments. Also, we
consider the behavior of constants in the inequality for the characteristic
function of a trigonometric analogue of the above-mentioned system when the
random variables are independent and uniformly distributed. Both inequalities
were derived earlier by the author from a multidimensional analogue of
Vinogradov's inequality for a trigonometric integral. As a byproduct the lower
bound for the spectrum of $A_kA_k'$ is obtained, where $A_k$ is the matrix of
coefficients of the first $k+1$ Chebyshev polynomials of first kind.