Market Mill Dependence Pattern in the Stock Market: Distribution   Geometry, Moments and Gaussization
  This paper continues a series of studies devoted to analysis of the bivariate probability distribution P(x,y) of two consecutive price increments x (push) and y (response) at intraday timescales for a group of stocks. Besides the asymmetry properties of P(x,y) such as Market Mill dependence patterns described in preceding paper [1], there are quite a few other interesting geometrical properties of this distribution discussed in the present paper, e.g. transformation of the shape of equiprobability lines upon growing distance from the origin of xy plane and approximate invariance of P(x,y) with respect to rotations at the multiples of $\pi/2$ around the origin of xy plane. The conditional probability distribution of response P(y|x) is found to be markedly non-gaussian at small magnitude of pushes and tending to more gauss-like behavior upon growing push magnitude. The volatility of P(y|,x) measured by the absolute value of the response shows linear dependence on the absolute value of the push, and the skewness of P(y|x) is shown to inherit a sign of the push. The conditional dynamics approach applied in this study is compared to regression models of AR-ARCH class. 
Andrei Leonidov, Vladimir Trainin, Alexander Zaitsev, Sergey Zaitsev