Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine.We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky). © Institute of Mathematical Statistics, 2013.

Mental hoop diaries: emotional memories of a college basketball game in rival fans.

The rivalry between the men's basketball teams of Duke University and the University of North Carolina-Chapel Hill (UNC) is one of the most storied traditions in college sports. A subculture of students at each university form social bonds with fellow fans, develop expertise in college basketball rules, team statistics, and individual players, and self-identify as a member of a fan group. The present study capitalized on the high personal investment of these fans and the strong affective tenor of a Duke-UNC basketball game to examine the neural correlates of emotional memory retrieval for a complex sporting event. Male fans watched a competitive, archived game in a social setting. During a subsequent functional magnetic resonance imaging session, participants viewed video clips depicting individual plays of the game that ended with the ball being released toward the basket. For each play, participants recalled whether or not the shot went into the basket. Hemodynamic signal changes time locked to correct memory decisions were analyzed as a function of emotional intensity and valence, according to the fan's perspective. Results showed intensity-modulated retrieval activity in midline cortical structures, sensorimotor cortex, the striatum, and the medial temporal lobe, including the amygdala. Positively valent memories specifically recruited processing in dorsal frontoparietal regions, and additional activity in the insula and medial temporal lobe for positively valent shots recalled with high confidence. This novel paradigm reveals how brain regions implicated in emotion, memory retrieval, visuomotor imagery, and social cognition contribute to the recollection of specific plays in the mind of a sports fan.