The parameter ⟨β⟩, measuring confinement efficiency, is one of the main figures of merit of a magnetic fusion device. Present-day limits to its increase are not imposed by disruptive MHD activity but rather by edge confinement degradation. The Large Helical Device heliotron has recently achieved record ⟨β⟩ values up to 5%. At the same time, recent fast camera experiments have revealed that this operational regime is characterized by the stochastic ejection of macroscopic structures in the form of filaments. In this work we apply suitable statistical tools to the analysis of the ejection sequence of such structures, as measured by the camera, and show that it possesses a self-similar, persistent character. Some self-similar properties of the stochastic process (namely the Hurst parameter and the waiting-time distribution) are studied and found to be dependent on the value of ⟨β⟩. These experimental results are interpreted in the light of previous theoretical works and the existence of self-organization is discussed.