Counting Extreme Points from Poisson Processes on a Half Line

Run a Poisson process to generate points on the positive vertical axis, so that the counting process looks like an increasing arc with random jagged edges (see Figure 1.2). The outermost Poisson points—the extreme points—are those that sit on the boundary of the counting process\u27 convex hull. How many extreme points are there? This thesis examines numerous approaches to this question with different styles of answers. Originally, the inspiration for this problem and the purpose of an answer was to guess a growth exponent for the extreme primes studied by McNew (2018), Tutaj (2018), and Pomerance (1979); from estimates here, one might guess 1/3. Upon exploration, the Poisson problem, certain results, and certain techniques herein have unmistakable ties to work by Groeneboom (2011) on a closely related problem about empirical distributions. In fact, the approach by Groeneboom (2011) would likely yield these 1/3 answers for our problem, as well (perhaps even with greater precision than we can provide), though we cannot say with complete certainty, since not all the details were laid out. Moreover, certain techniques here share features with the work by Groeneboom (2011), though the approach here begins from a slightly different point-by-point perspective. We also comment on these similarities and make use of this relationship. Aside from Poisson processes leading to growth exponent 1/3, we study other examples that have growth exponent 1 instead