Some inequalities for multivariate characteristic functions (Q1585962)

Some new lower and upper bounds are obtained for characteristic functions (cf's) of multivariate distributions that can be useful in limit theorems and stability problems. Many are multivariate extensions of results reported by the first author [J. Math. Sci., New York 84, No. 3, 1179-1189 (1997; Zbl 0915.60027)]. Notation: \({\mathbf X}\) is a random vector (rv) in \(R^m\), \(m\geq 1\), with distribution function (df) \(F({\mathbf x})\), density \(p({\mathbf x})\), cf \(f({\mathbf t})\) and covariance matrix \(\Sigma\). \(\text{R} f({\mathbf t})\) is the real part of \(f({\mathbf t})\). \(\langle{\mathbf x}, {\mathbf y}\rangle\) is the scalar product of vectors \({\mathbf x}\) and \({\mathbf y}\); \(\|{\mathbf x}\|=\sqrt{(\langle {\mathbf x},{\mathbf x}\rangle)}\). \(h(z)=2(1-\cos z)/z^2\) and \(k(z)=(2/z^2) \log(2/(2-z^2))\). Theorem 1. For \(u>0\) and any \(a>m/\sqrt \pi:\int_{\|{\mathbf x}\|\geq a/u}dF(x)\leq c(a)/u^m \int_{ \|{\mathbf t}\|\leq u}[1-Rf ({\mathbf t})] d{\mathbf t}\). Here \(c(a)= \Gamma(m/2 +1)/ \pi^{m/2}[1-m/a \sqrt\pi]\). Theorem 2. If \(\|{\mathbf X} \|\leq c< \infty\) and \(\sup p({\mathbf x})\leq a<\infty\), then \(|f({\mathbf t}) |\leq (\sin d)/d\) for \(\|{\mathbf t}\|\leq\pi/2c\) and \(|f({\mathbf t}) |\leq (\sin D)/D\) for \(\|{\mathbf t}\|>\pi/2c\). Here \(b=\pi^{(m-1)/2} c^{m-1}a/ \Gamma((m+1)/2)\), \(d=\|{\mathbf t}\|/2b\), \(D=\pi/4bc\). Theorem 3. If \(\|{\mathbf X}\|\leq c\) and \(a\in[0,\pi/4]\), then \(|f({\mathbf t}) |\leq 1-h(a){\mathbf t}\Sigma{\mathbf t}'/2\) for \(\|{\mathbf t}\|\leq a/c\). Theorem 4. \(|f({\mathbf t})|\geq 1-{\mathbf t}\Sigma {\mathbf t}'/2\) for all \({\mathbf t}\in R^m\). Theorem 5. \(|f({\mathbf t})|\geq\exp \{-k(a) {\mathbf t}\Sigma {\mathbf t}'/2\}\) for any \(a\in(0,\sqrt 2)\) and all \({\mathbf t}\in R^m\) such that \({\mathbf t}\Sigma {\mathbf t}'\leq a^2\). Theorem 6. Let \(g(z)\), \(z\geq 0\), be a non-negative, unbounded, increasing function. Let \(a\) and \(d\) be finite real numbers. If \(\sup p({\mathbf x})\leq a\), \(Eg(\|{\mathbf X}\|)=d\) and \(\delta\in (0,1]\), then \(|f({\mathbf t})\leq 1-(1-\delta)^2 \|{\mathbf t} \|^2/3 \pi^2b^2\) for \(\|{\mathbf t}\|\leq\pi/2c\) and \(|f({\mathbf t}) |\leq 1-(1- \delta)^3/ 12b^2c^2\) for \(\|{\mathbf t}\|> \pi/2c\). Here \(c= g^{-1}(d/ \delta)\) and \(b\) is as in Theorem 2. Theorem 7. If \(\sup p({\mathbf x})\leq a<\infty\), then \(|f({\mathbf t})|\leq 1-c\|{\mathbf t}\|^2\) for \(\sqrt{{\mathbf t}\Sigma{\mathbf t}'}\leq\pi/4\) and \(|f({\mathbf t})|\leq 1-c\pi^2 \|{\mathbf t}\|^2/ 16({\mathbf t}\Sigma {\mathbf t}')\) for \(\sqrt{{\mathbf t}\Sigma{\mathbf t}'}> \pi/4\). Here \(\sigma^2= E\|{\mathbf X}\|^2\) and \(c=9[ \Gamma((m+1)/2)]^2/2^{m+8} \pi^{m+1}a^2(\sigma^2)^{m-1}\).