Many things are similar in the eventual limit.

**under construction ⚠️**: I merged two notebooks here. The seams are showing.

We use asymptotic approximations all the time in statistics, most frequently in asymptotic pivots that motivate classical tests e.g. in classical hypothesis tests or an information penalty. We use the asymptotic delta method to motivate robust statistics, or infinite neural networks. There are various specialised mechanism; I am fond of the Stein methods. Also fun, Feynman-Kac formulae give us central limit theorems for all manner of weird processes.

There is much to be said on the various central limit theorems, but I will not be the one to say it right this minute, because this is a placeholder.

A convenient feature of M-estimation, and especially maximum likelihood estimation is simple behaviour of estimators in the asymptotic large-sample-size limit, which can give you, e.g. variance estimates, or motivate information criteria, or robust statistics, optimisation etc.

In the most celebrated and convenient cases case asymptotic bounds are about normally-distributed errors, and these are typically derived through *Local Asymptotic Normality* theorems.
A simple and general introduction is given in Andersen et al. (1997) page 594., which applies to both i.i.d. data and dependent_data in the form of
point processes.
For all that it is applied, it is still stringent.

In many nice distributions, central limit theorems lead (Asymptotically) to Gaussian distributions, and we can treat uncertainty in terms of transformations of Gaussians.

## Bayesian posteriors are kinda gaussian

The Bayesian large sample result of notes is the Bernstein–von Mises theorem, which provides some conitions under which the posterior distribution is asymptotically Gaussian.

## Particle filters are kinda gaussian

Long story. See (Bishop and Del Moral 2023, 2016; Cérou et al. 2005; Pierre Del Moral, Hu, and Wu 2011; Pierre Del Moral 2004; Pierre Del Moral and Doucet 2009; P. Del Moral, Kurtzmann, and Tugaut 2017).

## Fisher Information

Used in ML theory and kinda-sorta in robust estimation, and natural gradient methods. A matrix that tells is how much a new datum affects our parameter estimates. (It is related, I am told, to garden-variety Shannon information, and when that non-obvious fact is more clear to me I shall expand how precisely this is so.) 🏗

## Convolution Theorem

The unhelpfully-named convolution theorem of Hájek (1970) — is that related?

Suppose \(\hat{\theta}\) is an efficient estimator of \(\theta\) and \(\tilde{\theta}\) is another, not fully efficient, estimator. The convolution theorem says that, if you rule out stupid exceptions, asymptotically \(\tilde{\theta} = \hat{\theta} + \varepsilon\) where \(\varepsilon\) is pure noise, independent of \(\hat{\theta}.\)

The reason that’s almost obvious is that if it weren’t true, there would be some information about \(\theta\) in \(\tilde{\theta}-\hat{\theta}\), and you could use this information to get a better estimator than \(\hat{\theta}\), which (by assumption) can’t happen. The stupid exceptions are things like the Hodges superefficient estimator that do better at a few values of \(\hat{\theta}\) but much worse at neighbouring values.

## References

*Proceeding of the Second International Symposium on Information Theory*, edited by Petrovand F Caski, 199–213. Budapest: Akademiai Kiado.

*Biometrika*60 (2): 255–65.

*Statistical models based on counting processes*. Corr. 2. print. Springer series in statistics. New York, NY: Springer.

*Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)*39 (2): 101–23.

*Measure theory and probability theory*. New York: Springer.

*Stochastic Processes and Their Applications*, A Special Issue on the Occasion of the 2013 International Year of Statistics, 123 (7): 2475–99.

*An Introduction to Stein’s Method*. Vol. 4. Lecture Notes Series / Institute for Mathematical Sciences, National University of Singapore, v. 4. Singapore : Hackensack, N.J: Singapore University Press ; World Scientific.

*International Statistical Review / Revue Internationale de Statistique*62 (1): 133–65.

*Lectures on Empirical Processes: Theory and Statistical Applications*. European Mathematical Society.

*The Annals of Probability*14 (1): -336-342.

*The Annals of Probability*32 (1): 730–56.

*Bernoulli*1 (1/2): 17–39.

*SIAM Journal on Control and Optimization*55 (6): 4015–47.

*Mathematics of Control, Signals, and Systems*, May.

*Monte Carlo and Quasi-Monte Carlo Methods*, edited by Ronald Cools and Dirk Nuyens, 163:245–60. Springer Proceedings in Mathematics & Statistics. Cham: Springer International Publishing.

*Journal of the American Statistical Association*96 (455): 1022–30.

*Proceedings of the Winter Simulation Conference, 2005.*

*Statistics on Special Manifolds*, by Yasuko Chikuse, 174:187–230. New York, NY: Springer New York.

*Biometrika*96 (3): 529–44.

*Asymptotic Theory of Statistics and Probability*. Springer Texts in Statistics. New York: Springer New York.

*Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications*. 2004 edition. Latheronwheel, Caithness: Springer.

*On the Concentration Properties of Interacting Particle Processes*. Vol. 3. Now Publishers.

*SIAM Journal on Control and Optimization*55 (1): 119–55.

*Stochastic Processes and Their Applications*125 (4): 1195–1217.

*Advances in Applied Probability*8 (4): 712–36.

*The Annals of Mathematical Statistics*22 (3): -427-432.

*von Mises calculus for statistical functionals*. Lecture Notes in Statistics 19. New York: Springer.

*Annals of Probability*18 (2): 851–69.

*Journal of Statistical Planning and Inference*80 (1-2): 81–93.

*arXiv:1706.07180 [Cs, Math, Stat]*, June.

*Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete*14 (4): 323–30.

*Selected Works of C.C. Heyde*, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214–35. Selected Works in Probability and Statistics. Springer New York.

*The Annals of Mathematical Statistics*35 (1): 73–101.

*arXiv:1708.03625 [Stat]*, July.

*The Annals of Statistics*38 (3): 1478–1545.

*Limit Theorems for Stochastic Processes*. Vol. 288. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.

*Limit Theorems for Stochastic Processes*, edited by Jean Jacod and Albert N. Shiryaev, 1–63. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg.

*arXiv:1610.01353 [Math, Stat]*, October.

*arXiv:1211.4039 [Math]*, November.

*Biometrika*83 (4): 875–90.

*Journal of Statistical Planning and Inference*, C.R. Rao 80th Birthday Felicitation Volume, Part IV, 114 (1–2): 45–61.

*Biometrika*101 (1): 141–54.

*The Annals of Mathematical Statistics*41 (3): 802–28.

*Bernoulli*20 (4): 2020–38.

*The Annals of Statistics*4 (1): 51–67.

*arXiv:1802.00762 [Math]*, February.

*Annals of the Institute of Statistical Mathematics*30 (1): 243–61.

*Empirical Processes: Theory and Applications*. IMS.

*Electronic Journal of Statistics*12 (1): 890–940.

*Proceedings of the National Academy of Sciences of the United States of America*83 (3): 541–45.

*Concentration of Measure Inequalities in Information Theory, Communications, and Coding: Second Edition*. Now Publishers.

*Probability Surveys*8 (0): 210–93.

*arXiv:1409.2090 [Math, Stat]*, September.

*Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete*69 (3): 439–59.

*The Econometrics Journal*3 (2): 123–47.

*Journal of Applied Probability*19 (1): 221–28.

*Approximate Computation of Expectations*. Vol. 7. IMS.

*An Introduction to Matrix Concentration Inequalities*.

*Asymptotic statistics*. 1. paperback ed., 8. printing. Cambridge series in statistical and probabilistic mathematics. Cambridge: Cambridge Univ. Press.

## No comments yet. Why not leave one?