## Bernoulli

• Bernoulli
• Volume 20, Number 2 (2014), 623-644.

### Statistical convergence of Markov experiments to diffusion limits

#### Abstract

Assume that one observes the $k\text{th},2k\text{th},\ldots,nk\text{th}$ value of a Markov chain $X_{1,h},\ldots ,X_{nk,h}$. That means we assume that a high frequency Markov chain runs in the background on a very fine time grid but that it is only observed on a coarser grid. This asymptotics reflects a set up occurring in the high frequency statistical analysis for financial data where diffusion approximations are used only for coarser time scales. In this paper, we show that under appropriate conditions the $\mathrm{L}_{1}$-distance between the joint distribution of the Markov chain and the distribution of the discretized diffusion limit converges to zero. The result implies that the LeCam deficiency distance between the statistical Markov experiment and its diffusion limit converges to zero. This result can be applied to Euler approximations for the joint distribution of diffusions observed at points $\Delta,2\Delta,\ldots ,n\Delta$. The joint distribution can be approximated by generating Euler approximations at the points $\Delta k^{-1},2\Delta k^{-1},\ldots ,n\Delta$. Our result implies that under our regularity conditions the Euler approximation is consistent for $n\to\infty$ if $nk^{-2}\to0$.

#### Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 623-644.

Dates
First available in Project Euclid: 28 February 2014

https://projecteuclid.org/euclid.bj/1393594000

Digital Object Identifier
doi:10.3150/12-BEJ500

Mathematical Reviews number (MathSciNet)
MR3178512

Zentralblatt MATH identifier
1321.60010

#### Citation

Konakov, Valentin; Mammen, Enno; Woerner, Jeannette. Statistical convergence of Markov experiments to diffusion limits. Bernoulli 20 (2014), no. 2, 623--644. doi:10.3150/12-BEJ500. https://projecteuclid.org/euclid.bj/1393594000

#### References

• [1] Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 906–937.
• [2] Brown, L.D. and Low, M.G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
• [3] Brown, L.D., Wang, Y. and Zhao, L.H. (2003). On the statistical equivalence at suitable frequencies of GARCH and stochastic volatility models with the corresponding diffusion model. Statist. Sinica 13 993–1013. Statistical applications in financial econometrics.
• [4] Buchmann, B. and Müller, G. (2012). Limit experiments of GARCH. Bernoulli 18 64–99.
• [5] Dacunha-Castelle, D. and Florens-Zmirou, D. (1986). Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19 263–284.
• [6] Dalalyan, A. and Reiß, M. (2006). Asymptotic statistical equivalence for scalar ergodic diffusions. Probab. Theory Related Fields 134 248–282.
• [7] Dalalyan, A. and Reiß, M. (2007). Asymptotic statistical equivalence for ergodic diffusions: The multidimensional case. Probab. Theory Related Fields 137 25–47.
• [8] Duval, C. and Hoffmann, M. (2011). Statistical inference across time scales. Electron. J. Stat. 5 2004–2030.
• [9] Fischer, M. and Nappo, G. (2010). On the moments of the modulus of continuity of Itô processes. Stoch. Anal. Appl. 28 103–122.
• [10] Genon-Catalot, V., Laredo, C. and Nussbaum, M. (2002). Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Statist. 30 731–753. Dedicated to the memory of Lucien Le Cam.
• [11] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. Probability and Mathematical Statistics. New York: Academic Press [Harcourt Brace Jovanovich Publishers].
• [12] Konakov, V. and Mammen, E. (2009). Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains. Probab. Theory Related Fields 143 137–176.
• [13] Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535–543.
• [14] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
• [15] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772–802.
• [16] Wang, Y. (2002). Asymptotic nonequivalence of Garch models and diffusions. Ann. Statist. 30 754–783. Dedicated to the memory of Lucien Le Cam.