Summary
For a class of stationary independent increments processes we find a necessary and sufficient integral test on a function 0<h∈↑ for P[R∩(x, x+h(x))=∅ i.o. x↑∞]=1 and for P[R t∩(x, x+h(x))=∅ i.o. x↓0] =1, for all t>0, where R= range of X, R t = range of X up to epoch t.
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Erickson, K.B. Gaps in the range of nearly increasing processes with stationary independent increments. Z. Wahrscheinlichkeitstheorie verw Gebiete 62, 449–463 (1983). https://doi.org/10.1007/BF00534197
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DOI: https://doi.org/10.1007/BF00534197