Research

• Probability limit theorems (research papers [1-4], [13], [17,18], [20], [32], see below )
  My research in the field of probability limit theorems focuses on the convergence rate of distributions of sums of independent and identically distributed random variables (and random vectors) to stable distributions. The main achievement is the so called "non-uniform bound" for the convergence rate.
• Limit theorems for random processes (research papers [4-12], [14-16], [19], [27], [32], see below)
  Main results establish sufficient conditions for the central limit theorem to hold for stochastically continuous random processes in Skorokhod space D[0,1] of CADLAG (right continuous and having left limits) functions. The case of Gaussian limit process (with continuous sample paths) and the case of stochastically continuous stable limit process (with sample paths having no discontinuities of the second kind) are treated separately. Conditions are formulated in terms of moments (weak moments in case of stable limit) of increments of summands (that are stochastically continuous processes with sample paths in D[0,1]).
• Mathematical statistics: approximation of distributions of non-linear statistics (research papers [17,18], [20,21], [33], see below)
  A new approach is used to study the accuracy of the normal approximation of M estimators in the case of asymptotically convex score function, see [18]. This approach (based on the law of large numbers for random functions) allows to show a Berry-Esseen bound under optimal conditions (minimal smoothness condition on the score function and optimal moment conditions). Accuracy of the normal approximation of distributions selfnormalized sums is studied in [17], [20,21]. A general and precise Berry-Esseen type bound for Student's t statistic is shown in [17] for independent not identically distributed observations. One term Edgeworth expansion is established in [20,21] under the optimal conditions: finite third moment and non-lattice distribution of summands.
• Mathematical statistics: approximation of distributions of finite population statistics (research papers [22-26], [28-31], [34-37], see below)
  The papers [22,24, 30,31] develop Hoeffding's decomposition for symmetric statistics and show one term Edgeworth expansion for distributions of symmetric statistics. Empirical Edgeworth expansion (based on jackknife estimators of moments defining the expansion) is constructed in [28,29]. One term Edgeworth expansion and its empirical counterpart is constructed for Studentized finite population statistics in [34,35]. Stratified samples are considered in [36,37].

Current research

• normal approximation and Edgeworth expansions for distributions of symmetric statistics;
• resampling approximation in finite populations;
• approximations of distributions of combinatorial statistics (sub-graph count).

List of publications

Articles in journals / contributions to books

1. Bloznelis, M. (1988): Rate of convergence to a stable law in the space R^k. Lithuanian Math. J. 28, 21-29.
2. Bentkus, V. and Bloznelis, M. (1989): Nonuniform estimate of the rate of convergence in the CLT with stable limit distribution. Lithuanian Math. J. 29, 8-17.
3. Bloznelis, M. (1989): Nonuniform estimate of the rate of convergence to a stable law in the multidimensional central limit theorem. Lithuanian Math. J. 29, 97-109.
4. Bloznelis, M. (1989): Lower bound for the rate of convergence in the CLT in Hilbert space. Lithuanian Math. J. 29, 333-338.
5. Bloznelis, M. (1990): Distribution of the norm of a multidimensional Wiener process. Lithuanian Math. J. 30, 296-302.
6. Bloznelis, M. (1991): On the distribution of the norm for a multidimensional Brownian Bridge. Lithuanian Math. J. 31, 19-27.
7. Bloznelis, M. (1991): On the distribution of the sup-norm for a stable motion and the rate of convergence. In New Trends in Probab. and Statist. Vol I, Eds. Sazonov, V. and Shervashidze, T. (Utrecht: VSP), 73-77.
8. Bloznelis, M. (1991): Distribution of multidimensional Wiener prosess and Brownian bridge. In Functional and Stochastics analysis, Eds. Ulyanov, P.L. and Gnedenko, B.V. (Moscow: Moscow University Press), 40-41 (in Russian).
9. Bloznelis, M. and Paulauskas, V. (1993): On the central limit theorem in D[0,1]. Statistics & Probability Letters, 17, 105-111.
10. Bloznelis, M. and Paulauskas, V. (1993): The central limit theorem in the space D(0,1). I. Lithuanian Math. J. 33, 181-195.
11. Bloznelis, M. and Paulauskas, V. (1993): The central limit theorem in the space D(0,1). II. Lithuanian Math. J. 33, 307-323.
12. Bloznelis, M. and Paulauskas, V. (1993): Central limit theorem in Skorohod spaces and asymptotic strength distribution of fiber bundles. In Probability Theory and mathematical Statistics. Eds. Grigelionis, B., Kubilius, J., Pragarauskas, H. and Statulevicius, V. (Utrecht/Vilnius:VSP/TEV), 75-87.
13. Bloznelis, M. (1994): Refined nonuniform estimates of the rate of convergence to a stable law. J. Math. Sci. 72, 2839-2847.
14. Bloznelis, M. and Paulauskas, V. (1994): A note on the central limit theorem for stochastically continuous processes. Stochastic Proc. Appl. 53, 351-361.
15. Bloznelis, M. and Paulauskas, V. (1994): On the central limit theorem for multiparameter stochastic processes. In Probability in Banach spaces 9. Eds. Hoffmann-Jorgensen, J., Marcus, M. and Kuelbs, J. ( Birkhauser), 155-172. (preliminary version, pdf)
16. Bloznelis, M. (1996): Central limit theorem for stochastically continuous processes. Convergence to stable limit. J. Theoretical Probability, 9, 541--560. (preliminary version, pdf)
17. Bentkus, V., Bloznelis, M. and Götze, F. (1996): A Berry-Esseen bound for Student's statistic in the non-i.i.d. case. J. Theoretical Probability, 9, 765--796. (preliminary version, pdf)
18. Bentkus, V., Bloznelis, M. and Götze, F. (1997): A Berry-Esseen bound for M--Estimators. Scandinavian Journal of Statistics, 24, 485--502. (preliminary version, pdf)
19. Bloznelis, M. (1997): On the rate of normal approximation in D(0,1). Liet. Mat. Rink. 37, 280-294. (preliminary version, pdf)
20. Bloznelis, M. and Ra\v ckauskas, A. (1999): A Berry--Esseen bound for least squares estimators of regression parameters. Liet. Matem. Rink. 39, 1--8. (preliminary version, pdf)
21. Bloznelis, M. and Putter, H. (1999): One term Edgeworth expansion for Student's t statistic. Prob. Theory and Math. Stat. Proceedings of the seventh Vilnius Conference, B.Grigelionis et al. (Eds), VSP/TEV, 81--98. (preliminary version, pdf)
22. Bloznelis, M. and Göotze F. (1999): One term Edgeworth expansion for finite population U-statistics of degree two. Acta Applicandae Mathematicae, 58, 75--90.
23. Bloznelis, M. (1999): A Berry-Esseen bound for finite population Student's statistic. Ann. Probab. 27, 2089--2108. (preliminary version, pdf)
24. Bloznelis, M. and Götze , F. (2000): An Edgeworth expansion for finite population U-statistics. Bernoulli, 6, 729--760. (preliminary version, pdf)
25. Bloznelis, M. (2000): One- and two-term Edgeworth expansions for finite population sample mean. Exact results.I. Liet. Matem. Rink. 40, 277--294. (preliminary version, pdf)
26. Bloznelis, M. (2000): One- and two-term Edgeworth expansions for finite population sample mean. Exact results.II. Liet. Matem. Rink. 40, 430--443. (preliminary version, pdf)
27. Bloznelis, M. and Paulauskas, V. (2000): Central limit theorem in D[0,1]. In Skorokhod's Ideas in Probability Theory. Korolyuk, V., Portenko, N., Syta, H.(Eds.) Mathematics and its Applications. Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine, Vol 32, Kyiv, 99--110. (preliminary version, pdf)
28. Bloznelis, M. (2001): Empirical Edgeworth expansion for finite population statistics.I. Liet. Matem. Rink. 41, 154--171. (preliminary version, pdf)
29. Bloznelis, M. (2001): Empirical Edgeworth expansion for finite population statistics. II. Liet. Matem. Rink. 41, 263--276. (preliminary version, pdf)
30. Bloznelis, M. and Götze, F. (2001): Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Stat. 29, 899--917. (preliminary version, pdf)
31. Bloznelis, M. and Götze, F. (2002): An Edgeworth expansion for symmetric finite population statistics. Ann. Probab. 30, 1238--1265. (preliminary version, pdf)
32. Bloznelis, M. (2002): A note on the multivariate local limit theorem. Statistics and Probability Letters, 59, 227--233. (preliminary version, pdf)
33. Bloznelis, M. and Putter, H. (2002): Second order and bootstrap approximation to Student's t statistic. Teorija Verojatnostei i ee Primenenija, 47, No 2, 374--381. Reprinted in Theory of Probability and Its Applications, 47, 300--307. (preliminary version, pdf)
34. Bloznelis, M. (2003): An Edgeworth expansion for Studentized finite population statistics. Acta Applicandae Mathematicae 78, 51--60. (preliminary version, pdf)
35. Bloznelis, M. (2003): Edgeworth expansions for Studentized versions of symmetric finite population statistics. Liet. Matem. Rink. 43, 271--293. (preliminary version, pdf)
36. Bloznelis, M. (2003): Consistency and bias of jackknife variance estimator in stratified samples. Statistics 38, 489--504. (preliminary version, pdf)
37. Bloznelis, M. (2004): On combinatorial Hoeffding decomposition and asymptotic normality of sub-graph count statistics. In Mathematics and Computer Science III. Algorithms, Trees, Combinatorics and Probabilities. Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B.(Eds.) Trends in Mathematics, Birkhauser, 73--79. (preliminary version, pdf)
38. Bloznelis, M. (2005): Orthogonal decomposition of symmetric functions defined on random permutations. Combinatorics, Probability and Computing 14, 249--268. (preliminary version, pdf)
    
    

    Submitted

39. Bloznelis, M. (2005): Second Order and Resampling Approximation of Finite Population U-Statistics Based on Stratified Samples. (preliminary version, pdf)
40. Bloznelis, M. (2006): Bootstrap approximation to distributions of finite population U-statistics. (preliminary version, pdf)
    
    

    Other

    Bloznelis, M. (2004): Normal approximation for stratified samples. In Workshop on Survey Sampling Theory and Methodology. June 18-22, 2004, Tartu Estonia. Plikusas, A., Krapavickaitė, D., Kulldorff, G., Traat, I., Lapins, J. (Eds.) Statistical Office of Estonia, Tartu. 18--23. (preliminary version, pdf)