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On the limit law of a random walk conditioned to reach a high level. (English) Zbl 1225.60076

Summary: We consider a random walk with a negative drift and with a jump distribution which under Cramér’s change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally positive Lévy process conditioned not to overshoot level 1.

MSC:

60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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