Inequalities for the Probability Content of a Rotated Square and Related Convolutions

Hall, Richard L. ; Kanter, Marek ; Perlman, Michael D.

Ann. Probab., Tome 8 (1980) no. 6, p. 802-813 / Harvested from Project Euclid

Résumé

Let $(X_1, X_2)$ be independent $N(0, 1)$ variables and let $P(v_1, v_2) = P\lbrack(X_1, X_2) \in C + (v_1, v_2)\rbrack$, where $C$ is the square $\{|x_1| \leqslant a,|x_2| \leqslant a\}$. By demonstrating that $P\lbrack|X_i - v_i|\leqslant a\rbrack$ is $\log$ concave in $v^2_i$, the extrema of $P(v_1, v_2)$ on all circles $\{v^2_1 + v^2_2 = b^2\}$ are determined. The results are extended to determine the extrema of the probability of a cube in $R^n$. The proof is based on a log concavity-preserving property of the Laplace transforms.

Détail

Publié le : 1980-08-14
Classification:  Logarithmic concavity,  logarithmic convexity,  increasing failure rate,  decreasing failure rate,  Laplace transform,  convolution,  Gaussian density,  noncentral chi-squared distribution,  square,  cube,  Schur concavity,  26A51,  60D05,  60E05,  62H15