  | 
	
	
	 | 
	 | 
	
		 |  |  |  |  | 	 | 
	 | 
	 | 
	
		
	 | 
	 | 
	 | 
	 
	
 
 
	
	    
Convergence in Fractional Models and Applications	   
  
	 | 
  
 
	  
		 
			
			   
Corinne  Berzin, LabSad, Université Pierre Mendès-France José Rafael León, Universidad Central de Venezuelz 			 | 
		  
	   
		
  
		
			 
				
					   
					   Abstract 
	We consider a fractional Brownian motion with Hurst parameter strictly between 0 and 1. 
We are interested in the asymptotic behaviour of  functionals of the increments of
this and related processes and we propose several probabilistic and statistical applications.
				   
 
  
				 | 
			  
		   
   
Full text: PDF
  Pages: 326-370
  Published on: March 4, 2005
 
  
	 | 
 
 
                
                         
                                
                                          
                                           Bibliography 
        - Azaïs, Jean-Marc; Wschebor, Mario. Almost sure oscillation of certain random processes. Bernoulli 2 (1996), no. 3, 257-270. MR1416866 (98b:60077)
 - Banach, S. Sur les lignes rectifiables et les surfaces dont l'aire est finie. Fund. Math. 7 (1925), 225-237.
 - Berman, Simeon M. Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Statist. 41 (1970), 1260-1272. MR0272035 (42 #6916)
 - Berman, Simeon M. A central limit theorem for the renormalized self-intersection local time of a stationary vector Gaussian process. Ann. Probab. 20 (1992), no. 1, 61-81. MR1143412 (92m:60019)
 - Berzin, Corinne; León, José R.; Ortega, Joaquín. Level crossings and local time for regularized Gaussian processes. Probab. Math. Statist. 18 (1998), no. 1, Acta Univ. Wratislav. No. 2045, 39-81. MR1644037 (99i:60087)
 - Berzin-Joseph, Corinne; León, José R.; Ortega, Joaquín. Non-linear functionals of the Brownian bridge and some applications. Stochastic Process. Appl. 92 (2001), no. 1, 11-30. MR1815177 (2002d:60015)
 - Berzin-Joseph, C.; León, J. R. Weak convergence of the integrated number of level crossings to the local time for Wiener processes. Teor. Veroyatnost. i Primenen. 42 (1997), no. 4, 757-771; translation in Theory Probab. Appl. 42 (1997), no. 4, 568-579 MR1618738 (99e:60181)
 - Black, F.; Scholes, M. The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973), 637-654.
 - Breuer, Peter; Major, Péter. Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 (1983), no. 3, 425-441. MR0716933 (85d:60042)
 - Chambers, Daniel; Slud, Eric. Central limit theorems for nonlinear functionals of stationary Gaussian processes. Probab. Theory Related Fields 80 (1989), no. 3, 323-346. MR0976529 (90a:60037)
 - Cutland, Nigel J.; Kopp, P. Ekkehard; Willinger, Walter. Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model. Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), 327-351, Progr. Probab., 36, Birkhäuser, Basel, 1995. MR1360285
 - Dobrushin, R. L.; Major, P. Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27-52. MR0550122 (81i:60019)
 - Doukhan, Paul; Massart, Pascal; Rio, Emmanuel. The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 1, 63-82. MR1262892 (95b:60030)
 - Dynkin, E. B. Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 (1988), no. 1, 1-57. MR0920254 (89f:60053)
 - Ho, Hwai Chung; Sun, Tze Chien. Limiting distributions of nonlinear vector functions of stationary Gaussian processes. Ann. Probab. 18 (1990), no. 3, 1159-1173. MR1062063 (92c:60031)
 - Hunt, G. A. Random Fourier transforms. Trans. Amer. Math. Soc. 71, (1951). 38-69. MR0051340 (14,465b)
 - Kac, M. On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, (1943). 314-320. MR0007812 (4,196d)
 - Lin, S. J. Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep. 55 (1995), no. 1-2, 121-140. MR1382288 (97j:60095)
 - Lyons, Terry. Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett. 1 (1994), no. 4, 451-464. MR1302388 (96b:60150)
 - Mandelbrot, Benoit B.; Van Ness, John W. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968), 422-437. MR0242239 (39 #3572)
 - Perera, Gonzalo; Wschebor, Mario. Crossings and occupation measures for a class of semimartingales. Ann. Probab. 26 (1998), no. 1, 253-266. MR1617048 (99k:60064)
 - Pontryagin, L. Ordinary differential equations. Adiwes International Series in Mathematics. Addison-Wesley Publishing Company, London-Paris. Translated from the Russian by L.Kacinskas and W.B.Counts. (1962).
 - Rio, Emmanuel. About the Lindeberg method for strongly mixing sequences. ESAIM Probab. Statist. 1 (1995/97), 35-61 (electronic). MR1382517 (97d:60046)
 - Taqqu, Murad S. Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 3, 203-238. MR0471045 (57 #10786)
   
                                   
 
  
                                 | 
                          
                   
	  
 
 
 
 | 
		
			
 
 
 
 
 
 
 
 
  
			
			
			
			 
		 | 
		
	| 
	 | 
	
    	 
    	
  
     | 
     | 
 
	 | 
	
		 |  |  |  |  | 
 
 Electronic Journal of Probability.   ISSN: 1083-6489 	 | 
	 |