統計数学セミナー
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Seminar on Probability and Statistics Wednesday April 15 2009 Tokyo 128 4:20-5:30 pm
Estimating the successive Blumenthal-Getoor indices for a discretely observed process
Jean JACOD Universite Paris VI Abstract Letting F be a Levy measure whose "tail" $F ([-x, x])$
admits an expansion $\sigma_{i\ge 1} a_i/x^\beta$ as $x \rightarrow 0$,
we call $\beta_1 > \beta_2 >...$ the successive Blumenthal-Getoor indices,
since $\beta_1$ is in this case the usual Blumenthal-Getoor index. This notion
may be extended to more general semimartingale. We propose here a method
to estimate the $\beta_i$'s and the coefficients $a_i$'s, or rather their
extension for semimartingales, when the underlying semimartingale $X$ is
observed at discrete times, on fixed time interval. The asymptotic is when
the time-lag goes to $0$. It is then possible to construct consistent estimators
for $\beta_i$ and $a_i$ for those $i$'s such that $\beta_i > \beta_1 /2$,
whereas it is impossible to do so (even when $X$ is a Levy process) for those
$i$'s such that $\beta_i < \beta_1 /2$. On the other hand, a central limit
theorem for $\beta_1$ is available only when $\beta_i < \beta_1 /2$:
consequently, when we can actually consistently estimate some $\beta_i$'s
besides $\beta_1$ , then no central limit theorem can hold, and correlatively
the rates of convergence become quite slow (although one know them explicitly):
so the results have some theoretical interest in the sense that they set up bounds
on what is actually possible to achieve, but the practical applications are probably quite thin.
(joint with Yacine Ait-Sahalia) |
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