Sampling Theorem --- Probability Analysis of Infinite Sequence Paths with Recursive Pivots

The probabilistic properties of paths of infinite random sequences constitute an important research direction within the fundamental theory of modern probability theory. Based on the recursive pivot selection mechanism of the quicksort algorithm, this paper constructs a novel probabilistic path model for recursive sampling and proposes and rigorously proves the Sampling Theorem. By constructing strictly monotonically decreasing probability sequences and performing squeeze inequality analysis, this paper systematically proves that, for both sampling with replacement and sampling without replacement, the probability of recursively selected paths asymptotically tends to 0 as the scale of the sampling system goes to infinity; and the probability of such paths remains strictly greater than 0 at all times. This study clarifies logical ambiguities existing in classical probability theory, rigorously distinguishes the asymptotic trend of infinite probability sequences from probability values, and improves the characterization of probabilities for infinite recursive random paths. The complete derivation framework corresponding to this paper has long been fully established, and this represents its first formal academic archival release.This PDF contains an appendix with a Chinese version of the manuscript.