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Total variation distance for discretely observed Lévy processes: A Gaussian approximation of the small jumps
- Source :
- Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. 57
- Publication Year :
- 2021
- Publisher :
- Institute of Mathematical Statistics, 2021.
-
Abstract
- It is common practice to treat small jumps of L\'evy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of L\'evy processes with infinite L\'evy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a L\'evy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of L\'evy processes are provided. The theory is illustrated by concrete examples.<br />Comment: Important and necessary changes have been made in this new version, this version supersedes version 1
- Subjects :
- Statistics and Probability
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Gaussian
Gaussian approximation
Mathematics - Statistics Theory
01 natural sciences
Measure (mathematics)
Lévy process
010104 statistics & probability
Total variation
symbols.namesake
Total variation distance
Statistical physics
0101 mathematics
Statistical hypothesis testing
Mathematics
[STAT.TH] Statistics [stat]/Statistics Theory [stat.TH]
010102 general mathematics
Statistical model
[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
60G51, 62M99 (Primary), 60E99 (Secondary)
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Distribution (mathematics)
Lévy processes
Small jumps
Metric (mathematics)
symbols
Statistics, Probability and Uncertainty
Statistical test
Mathematics - Probability
Subjects
Details
- ISSN :
- 02460203
- Volume :
- 57
- Database :
- OpenAIRE
- Journal :
- Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
- Accession number :
- edsair.doi.dedup.....be0306aeb303f1a2cfd8a83bd9dd4274