Let (μα) be a net of Radon sub-probability measures on ℝ, and (tα) be a net in ]0, 1] converging to 0. Assuming that the generalized log-moment generating function L(λ) exists for all λ in a nonempty open interval G, we give conditions on the left or right derivatives of L|G, implying a vague (and thus narrow when 0 ∈ G large deviation principle. The rate function (which can be nonconvex) is obtained as an abstract Legendre–Fenchel transform. This allows us to strengthen the Gärtner–Ellis theorem by weakening the essential smoothness assumption. A related question of R. S. Ellis is solved.
@article{CML_2009__1_2_181_0,
author = {Comman, Henri},
title = {Differentiability-free conditions on the free-energy function implying large deviations},
journal = {Confluentes Mathematici},
pages = {181--196},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {1},
number = {2},
year = {2009},
doi = {10.1142/S1793744209000079},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S1793744209000079/}
}
TY - JOUR AU - Comman, Henri TI - Differentiability-free conditions on the free-energy function implying large deviations JO - Confluentes Mathematici PY - 2009 SP - 181 EP - 196 VL - 1 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744209000079/ DO - 10.1142/S1793744209000079 LA - en ID - CML_2009__1_2_181_0 ER -
%0 Journal Article %A Comman, Henri %T Differentiability-free conditions on the free-energy function implying large deviations %J Confluentes Mathematici %D 2009 %P 181-196 %V 1 %N 2 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744209000079/ %R 10.1142/S1793744209000079 %G en %F CML_2009__1_2_181_0
Comman, Henri. Differentiability-free conditions on the free-energy function implying large deviations. Confluentes Mathematici, Tome 1 (2009) no. 2, pp. 181-196. doi : 10.1142/S1793744209000079. http://www.numdam.org/articles/10.1142/S1793744209000079/
[1] H. Comman, J. Theor. Probab. 18, 187 (2005), DOI: 10.1007/s10959-004-2594-2 .
[2] H. Comman, Trans. Amer. Math. Soc. 355, 2905 (2003), DOI: 10.1090/S0002-9947-03-03274-4 .
[3] A. Dembo and O. Zeitouni , Large Deviations Techniques and Applications , 2nd edn. ( Springer-Verlag , 1998 ) .
[4] R. S. Ellis, Scand. Actuarial J. 1, 97 (1995).
[5] R. S. Ellis , Entropy, Large Deviations, and Statistical Mechanics ( Springer-Verlag , 1985 ) .
[6] R. S. Ellis, Ann. Probab. 12, 1 (1984), DOI: 10.1214/aop/1176993370 .
[7] J. Gärtner, Th. Probab. Appl. 22, 24 (1977).
[8] G. L. O’Brien, Stoch. Process. Appl. 57, 1 (1995), DOI: 10.1016/0304-4149(95)00007-T .
[9] G. L. O’Brien and W. Vervaat, Stable Processes and Related Topics, eds. S. Cambanis, G. Samorodniski and M. S. Taqqu (Birkhäuser, 1991) pp. 43-83.
[10] R. T. Rockafeller , Convex Analysis ( Princeton Univ. Press , 1970 ) .
Cité par Sources :
