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A differential equation related to the $l^{p}$ -norms

Jacek Bojarski; Tomasz Małolepszy; Janusz Matkowski — 2011

Annales Polonici Mathematici

Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^{p}$ -distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

Band copulas as spectral measures for two-dimensional stable random vectors

Jacek Bojarski; Jolanta K. Misiewicz — 2003

Discussiones Mathematicae Probability and Statistics

In this paper, we study basic properties of symmetric stable random vectors for which the spectral measure is a copula, i.e., a distribution having uniformly distributed marginals.

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A differential equation related to the l p -norms

Band copulas as spectral measures for two-dimensional stable random vectors

A differential equation related to the $l^{p}$ -norms