Summary
We consider increasing processes {X(t)∶t≧0} of classL, that is, increasing self-similar processes with inswpendent increments. Leth(t) be an increasing positive function on (0,∞) withh(0+)=0 andh(∞)=∞. By virtue of the zero-one laws, there existsc (resp.C) ∈[0,∞] such that lim inf (resp. lim sup)X(t)/h(t)=c (resp.C) a.s. both ast tends to 0 and ast tends to ∞. We decide a necessary and sufficient condition for the existence ofh(t) withc orC=1 and explicitly constructh(t) in caseh(t) exists withc orC=1. Moreover, we give a criterion to classify functionsh(t) withc (orC)=0 andh(t) withc (orC)=∞ in caseh(t) does not exist withc (orC)=1.
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Watanabe, T. Sample function behavior of increasing processes of class L. Probab. Th. Rel. Fields 104, 349–374 (1996). https://doi.org/10.1007/BF01213685
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DOI: https://doi.org/10.1007/BF01213685