## Abstract

For nonnegative integers r, s, let ^{(} ^{r} ^{,} ^{s} ^{)}X_{t} be the Lévy process X_{t} with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let ^{(} ^{r} ^{)}X~ _{t} be X_{t} with the r largest jumps in modulus up till time t deleted. Let a_{t}∈ R and b_{t}> 0 be non-stochastic functions in t. We show that the tightness of (^{(} ^{r} ^{,} ^{s} ^{)}X_{t}- a_{t}) / b_{t} or (^{(} ^{r} ^{)}X~ _{t}- a_{t}) / b_{t} as t↓ 0 implies the tightness of all normed ordered jumps, and hence the tightness of the untrimmed process (X_{t}- a_{t}) / b_{t} at 0. We use this to deduce that the trimmed process (^{(} ^{r} ^{,} ^{s} ^{)}X_{t}- a_{t}) / b_{t} or (^{(} ^{r} ^{)}X~ _{t}- a_{t}) / b_{t} converges to N(0, 1) or to a degenerate distribution as t↓ 0 if and only if (X_{t}- a_{t}) / b_{t} converges to N(0, 1) or to the same degenerate distribution, as t↓ 0.

Original language | English |
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Pages (from-to) | 675-699 |

Number of pages | 25 |

Journal | Journal of Theoretical Probability |

Volume | 30 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2017 |