THE LIMITING LOG GAUSSIANITY FOR AN EVOLVING BINOMIAL RANDOM FIELD

- Journal title : Communications of the Korean Mathematical Society
- Volume 25, Issue 2, 2010, pp.291-301
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2010.25.2.291

Title & Authors

THE LIMITING LOG GAUSSIANITY FOR AN EVOLVING BINOMIAL RANDOM FIELD

Kim, Sung-Yeun; Kim, Won-Bae; Bae, Jong-Sig;

Kim, Sung-Yeun; Kim, Won-Bae; Bae, Jong-Sig;

Abstract

This paper consists of two main parts. Firstly, we introduce an evolving binomial process from a binomial stock model and consider various types of limiting behavior of the logarithm of the evolving binomial process. Among others we find that the logarithm of the binomial process converges weakly to a Gaussian process. Secondly, we provide new approaches for proving the limit theorems for an integral process motivated by the evolving binomial process. We provide a new proof for the uniform strong LLN for the integral process. We also provide a simple proof of the functional CLT by using a restriction of Bernstein inequality and a restricted chaining argument. We apply the functional CLT to derive the LIL for the IID random variables from that for Gaussian.

Keywords

evolving binomial process;limiting log Gaussian property;uniform LLN;functional CLT;LIL;

Language

English

Cited by

References

1.

R. M. Dudley and W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrsch. Verw. Gebiete 62 (1983), no. 4, 509-552.

2.

E. Gine and J. Zinn, Some limit theorems for empirical processes, Ann. Probab. 12 (1984), no. 4, 929-998.

3.

I. Karatzas, Lectures on the Mathematics of Finance, CRM Monograph Series, 8. American Mathematical Society, Providence, RI, 1997.

4.

A. F. Karr, Probability, Springer-Verlag, New York, 1993.

5.

M. Ossiander, A central limit theorem under metric entropy with $L_2$ bracketing, Ann. Probab. 15 (1987), no. 3, 897-919.

6.

D. Pollard, Convergence of Stochastic Processes, Springer series in Statistics, Springer-Verlag, New York, 1984.

7.

D. Pollard, Empirical Processes, Regional conference series in Probability and Statistics 2, Inst. Math. Statist., Hayward, CA, 1990.

8.

A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer series in Statistics, Springer-Verlag, New York, 1996.