The Annals of Probability

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

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

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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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Ann. Probab., Volume 8, Number 4 (1980), 802-813.

First available in Project Euclid: 19 April 2007

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Primary: 26A51: Convexity, generalizations
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60E05: Distributions: general theory 62H15: Hypothesis testing

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


Hall, Richard L.; Kanter, Marek; Perlman, Michael D. Inequalities for the Probability Content of a Rotated Square and Related Convolutions. Ann. Probab. 8 (1980), no. 4, 802--813. doi:10.1214/aop/1176994667.

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