## Bernoulli

• Bernoulli
• Volume 24, Number 2 (2018), 1033-1052.

### Functional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblies

Koji Tsukuda

#### Abstract

Functional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblies are presented. The random elements argued in this paper are viewed as elements taking values in $L^{2}(0,1)$ whereas the Skorokhod space is argued as a framework of weak convergences in functional central limit theorems for random combinatorial structures in the literature. It enables us to treat other standardized random processes which converge weakly to a corresponding Gaussian process with additional assumptions.

#### Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 1033-1052.

Dates
First available in Project Euclid: 21 September 2017

https://projecteuclid.org/euclid.bj/1505980888

Digital Object Identifier
doi:10.3150/16-BEJ847

Mathematical Reviews number (MathSciNet)
MR3706786

Zentralblatt MATH identifier
06778357

#### Citation

Tsukuda, Koji. Functional central limit theorems in $L^{2}(0,1)$ for logarithmic combinatorial assemblies. Bernoulli 24 (2018), no. 2, 1033--1052. doi:10.3150/16-BEJ847. https://projecteuclid.org/euclid.bj/1505980888

#### References

• [1] Arratia, R., Barbour, A.D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519–535.
• [2] Arratia, R., Barbour, A.D. and Tavaré, S. (2000). Limits of logarithmic combinatorial structures. Ann. Probab. 28 1620–1644.
• [3] Arratia, R., Stark, D. and Tavaré, S. (1995). Total variation asymptotics for Poisson process approximations of logarithmic combinatorial assemblies. Ann. Probab. 23 1347–1388.
• [4] Arratia, R. and Tavaré, S. (1992). Limit theorems for combinatorial structures via discrete process approximations. Random Structures Algorithms 3 321–345.
• [5] Arratia, R. and Tavaré, S. (1994). Independent process approximations for random combinatorial structures. Adv. Math. 104 90–154.
• [6] DeLaurentis, J.M. and Pittel, B.G. (1985). Random permutations and Brownian motion. Pacific J. Math. 119 287–301.
• [7] Donnelly, P.J., Ewens, W.J. and Padmadisastra, S. (1991). Functionals of random mappings: Exact and asymptotic results. Adv. in Appl. Probab. 23 437–455.
• [8] Ewens, W.J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 87–112; erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376.
• [9] Flajolet, P. and Soria, M. (1990). Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory Ser. A 53 165–182.
• [10] Hansen, J.C. (1989). A functional central limit theorem for random mappings. Ann. Probab. 17 317–332. Correction: Ann. Probab. 19 (1991) 1393–1396.
• [11] Hansen, J.C. (1990). A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27 28–43.
• [12] Prohorov, Yu.V. (1956). Convergence of random processes and limit theorems in probability theory. Teor. Veroyatn. Primen. 1 177–238.
• [13] Teicher, H. (1955). An inequality on Poisson probabilities. Ann. Math. Stat. 26 147–149.
• [14] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.
• [15] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.