TY - THES
AU - Padayachee, Krishna
PY - 1981
TI - Some applications of Choquet’s integral representation theorem to probability
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - In this thesis we look at the applications of Choquet's integral representation to probability theory. Applications of Choquet's theorem are given to obtain a representation of superharmonic functions on the Martin Boundary, a representation theorem for invariant measures with respect to a family of transformations T and finally to symmetric measures on a product space. In order to obtain the desired representation theorem in the above mentioned/applications we need to consider an appropriate topology on the spaces. In the case of the Martin boundary our underlying space is R[sup ∞] equipped with the product topology. The set of all superharmonic functions is shown to be a compact convex metrizable subset of R[sup ∞]. Furthermore the extreme points are isolated and they turn out to be the minimal harmonic functions. With regards to the other two applications we consider the space of measures on an appropriate topological space. The probability measures invariant with respect to a family of transformations T form a Compact convex set in the weak-star topology and the extreme points are the ergodic measures. In the case of the symmetric measures on the product space the symmetric probability measures form a compact convex set in the weak-star topology and the extreme points are the product probability measures.
N2 - In this thesis we look at the applications of Choquet's integral representation to probability theory. Applications of Choquet's theorem are given to obtain a representation of superharmonic functions on the Martin Boundary, a representation theorem for invariant measures with respect to a family of transformations T and finally to symmetric measures on a product space. In order to obtain the desired representation theorem in the above mentioned/applications we need to consider an appropriate topology on the spaces. In the case of the Martin boundary our underlying space is R[sup ∞] equipped with the product topology. The set of all superharmonic functions is shown to be a compact convex metrizable subset of R[sup ∞]. Furthermore the extreme points are isolated and they turn out to be the minimal harmonic functions. With regards to the other two applications we consider the space of measures on an appropriate topological space. The probability measures invariant with respect to a family of transformations T form a Compact convex set in the weak-star topology and the extreme points are the ergodic measures. In the case of the symmetric measures on the product space the symmetric probability measures form a compact convex set in the weak-star topology and the extreme points are the product probability measures.
UR - https://open.library.ubc.ca/collections/831/items/1.0080347
ER - End of Reference