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August, 1980 Inequalities for the Probability Content of a Rotated Square and Related Convolutions
Richard L. Hall, Marek Kanter, Michael D. Perlman
Ann. Probab. 8(4): 802-813 (August, 1980). DOI: 10.1214/aop/1176994667


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.


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Richard L. Hall. Marek Kanter. Michael D. Perlman. "Inequalities for the Probability Content of a Rotated Square and Related Convolutions." Ann. Probab. 8 (4) 802 - 813, August, 1980.


Published: August, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0452.60024
MathSciNet: MR577317
Digital Object Identifier: 10.1214/aop/1176994667

Primary: 26A51‎
Secondary: 60D05 , 60E05 , 62H15

Keywords: convolution , cube , decreasing failure rate , Gaussian density , increasing failure rate , Laplace transform , Logarithmic concavity , logarithmic convexity , noncentral chi-squared distribution , Schur concavity , square

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 4 • August, 1980
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