Persistence probabilities for fractionally integrated fractional Brownian noise (2509.10265v1)

Abstract: The main object the study is fractionally integrated fractional Brownian noise, I(t/a,H) where a>0 is the multiplicity(not necessarily an integer) of integration, and H is the Hurst parameter . The subject of the analysis is the persistence exponent e(a,H) that determines the power-law asymptotic of probability that the process will not exceed a fixit positive level in a growing time interval (0,T). In the important cases (a=1,H) and (a=2,H=1/2) these exponents are well known. To understand the problematic exponents e(2,H), we consider the (a,H) parameters from the maximum (for the task) area G= (a+H>1,a>0,0<H<1) ) . We prove the decrease of the exponents with increasing a and describe their behavior near the boundary of G, including infinity. The discovered identity of the exponents with the parameters (a,H) and (a+2H-1,1-H) actually refutes the long-standing hypothesis that e(2,H)=H(1-H). Our results are based on well known the continuity lemma for the persistence exponents and on a generalization of Slepian's lemma for a family of Gaussian processes smoothly dependent on a parameter.