On the Limit Law of a Random Walk Conditioned to Reach a High Level

We consider a random walk with a negative drift and with a jump distribution which under Cram'er’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 L'evy %-Khinchin process conditioned not to overshoot level one.