Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

To recover the topology of a manifold in presence heavy tailed or exponentially decaying noise, one must understand behavior geometric complexes whose points lie tail these noise distributions. This study advances this line inquiry, and demonstrates functional strong laws large numbers for Euler characteristic process random formed by outside an expanding ball \(\mathbb {R}^{d}\). When are drawn from distribution with regularly varying tail, grows at rate, scaled converges uniformly almost surely to smooth function. logarithmically, another function same sense. All limit theorems take place when inside densely distributed, so that simplex counts all dimensions contribute process.