a.k.a. “Fancy ARIMA”.

Classically, you estimate statistics from many i.i.d. realisations from a presumed generating process.

What if your data are realisations of sequentially dependent time series? How do you estimate parameters from a single time series realisation?

By being a flashy quant!

Bonus points: How do you do this with *many* time series, whose parameters themselves have a distribution you wish to estimate?

See Mark Podolskij who explains “high frequency asymptotics” well. I think that the original framework is due to Jacod. (i.e. when you don’t have an asymptotic limit in number of data points, but in how densely you sample a single time series.)

This feels contrived for me, but it is probably interesting if you are not working with a multivariate Brownian motion, but a rather general Lévy process or something with interesting jumps AND continuous movement, and can sample with arbitrary density but not arbitrarily long. AFAICT this is little outside finance.

## Refs

- BaSø94: (1994) A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes.
*International Statistical Review / Revue Internationale de Statistique*, 62(1), 133–165. DOI - HeSe10: (2010) Estimation Theory for Growth and Immigration Rates in a Multiplicative Process. In Selected Works of C.C. Heyde (pp. 214–235). Springer New York
- DuPo15: (2015) High-frequency asymptotics for path-dependent functionals of Itô semimartingales.
*Stochastic Processes and Their Applications*, 125(4), 1195–1217. DOI - JaPV10: (2010) Limit theorems for moving averages of discretized processes plus noise.
*The Annals of Statistics*, 38(3), 1478–1545. DOI - BiSø95: (1995) Martingale Estimation Functions for Discretely Observed Diffusion Processes.
*Bernoulli*, 1(1/2), 17–39. DOI - Feig76: (1976) Maximum Likelihood Estimation for Continuous-Time Stochastic Processes.
*Advances in Applied Probability*, 8(4), 712–736. DOI - Jaco97: (1997) On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités XXXI (pp. 232–246). Springer Berlin Heidelberg
- HePo14: (2014) On spectral distribution of high dimensional covariation matrices.
*ArXiv:1410.6764 [Math]*. - PoVe10: (2010) Understanding limit theorems for semimartingales: a short survey: Limit theorems for semimartingales.
*Statistica Neerlandica*, 64(3), 329–351. DOI