On  
conditional invariance principle
for random walks

O. Hryniv

Functional central limit theorem is proved for a %certain stochastic process constructed from a one-dimensional random walk with fixed endpoints of its trajectories. The limiting Gaussian measure corresponds to a Brownian bridge with orientation dependent parameters.

1. Akutsu, Y., Akutsu, N. Relationship between the anisotropic interface tension, the scaled interface width and the equilibrium shape in two dimensions // J. Phys. A: Math. Gen. V. 19 p. 2813--2820--1986

2. Billingsley, P. Convergence of probability measures , John Wiley & Sons –1968

3. DeConinck, J., Ruiz, J. Fluctuations of interfaces and anisotropy// J. Phys. A: Math. Gen. V.21 p.147–153 –1988

4. Dobrushin, R., Hryniv, O. Fluctuations of shapes of large areas under paths of random walks// Probab. Theory Relat. Fields V.105 p. 423–458 –1996

5. Dobrushin, R., Hryniv, O. Fluctuations of the Phase Boundary in the 2D Ising Ferromagnet// Commun. Math. Phys. V.189 p. 395–445–1997

6. Dobrushin, R., Kotecký, R., Shlosman, S. Wulff Construction: a Global Shape from Local Interaction. (Translations of mathematical monographs, 104.) , Providence, R.I.: Amer. Math. Soc. – 1992

7. Dobrushin, R., Shlosman, S. Large and Moderate Deviations in the Ising Model. Probability Contributions to Statistical Mechanics (Dobrushin R.L. ed.) (Advances in Soviet Mathematics, 20.) p.91–220, Providence, R.I.: Amer. Math. Soc.– 1994

8. Durrett, R. Conditioned Limit Theorems for Some Null Recurrent Markov Processes// Ann. Probab. V.6 p. 798–828 –1978

9. Durrett, R., Iglehart, D. L., and Miller, D. R. Weak Convergence to Brownian Meander and Brownian Excursion// Ann. Probab. V.5 p. 117–129–1977

10. Gnedenko, B. V. The theory of probability

11. Hryniv, O. On local behaviour of the phase separation line in the 2D Ising model Preprint ESI. 425., 1–21 (1997). // Probab. Theory Relat. Fields V.110 –1998

12. Iglehart, D. L. Functional Central Limit Theorem for Random Walks Conditioned to Stay Positive// Ann. Probab. V.2

13. Kaigh, W. D. An Invariance Principle for Random Walk Conditioned by a Late Return to Zero// Ann. Probab. V.4

14. Liggett, T. M. An Invariance Principle for Conditioned Sums of Independent Random Variables // J. Math. Mech. V.18

15. Sinai, Ya. G. Distributions of Some Functionals of Integrals of Random Walk// Theor. Math. Phys. V.90 p. 323–353 –1992

	On conditional invariance principle for random walks
Author	O. Hryniv Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine
Abstract	Functional central limit theorem is proved for a %certain stochastic process constructed from a one-dimensional random walk with fixed endpoints of its trajectories. The limiting Gaussian measure corresponds to a Brownian bridge with orientation dependent parameters.
Keywords	functional central limit theorem, stochastic process, one-dimensional random walk, fixed endpoints, limiting Gaussian measure, Brownian bridge
DOI	doi:10.30970/ms.9.1.102-109
Reference	1. Akutsu, Y., Akutsu, N. Relationship between the anisotropic interface tension, the scaled interface width and the equilibrium shape in two dimensions // J. Phys. A: Math. Gen. V. 19 p. 2813--2820--1986 2. Billingsley, P. Convergence of probability measures , John Wiley & Sons –1968 3. DeConinck, J., Ruiz, J. Fluctuations of interfaces and anisotropy// J. Phys. A: Math. Gen. V.21 p.147–153 –1988 4. Dobrushin, R., Hryniv, O. Fluctuations of shapes of large areas under paths of random walks// Probab. Theory Relat. Fields V.105 p. 423–458 –1996 5. Dobrushin, R., Hryniv, O. Fluctuations of the Phase Boundary in the 2D Ising Ferromagnet// Commun. Math. Phys. V.189 p. 395–445–1997 6. Dobrushin, R., Kotecký, R., Shlosman, S. Wulff Construction: a Global Shape from Local Interaction. (Translations of mathematical monographs, 104.) , Providence, R.I.: Amer. Math. Soc. – 1992 7. Dobrushin, R., Shlosman, S. Large and Moderate Deviations in the Ising Model. Probability Contributions to Statistical Mechanics (Dobrushin R.L. ed.) (Advances in Soviet Mathematics, 20.) p.91–220, Providence, R.I.: Amer. Math. Soc.– 1994 8. Durrett, R. Conditioned Limit Theorems for Some Null Recurrent Markov Processes// Ann. Probab. V.6 p. 798–828 –1978 9. Durrett, R., Iglehart, D. L., and Miller, D. R. Weak Convergence to Brownian Meander and Brownian Excursion// Ann. Probab. V.5 p. 117–129–1977 10. Gnedenko, B. V. The theory of probability 11. Hryniv, O. On local behaviour of the phase separation line in the 2D Ising model Preprint ESI. 425., 1–21 (1997). // Probab. Theory Relat. Fields V.110 –1998 12. Iglehart, D. L. Functional Central Limit Theorem for Random Walks Conditioned to Stay Positive// Ann. Probab. V.2 13. Kaigh, W. D. An Invariance Principle for Random Walk Conditioned by a Late Return to Zero// Ann. Probab. V.4 14. Liggett, T. M. An Invariance Principle for Conditioned Sums of Independent Random Variables // J. Math. Mech. V.18 15. Sinai, Ya. G. Distributions of Some Functionals of Integrals of Random Walk// Theor. Math. Phys. V.90 p. 323–353 –1992
Pages	102-109
Volume	9
Issue	1
Year	1998
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On conditional invariance principle for random walks