A. Definition, 15 (Limit of a stationary transfinite sequence) We say that the sequence (X ? ) ? ?µ is stationary if and only if ?? ? µ : ?? ? µ ? ? ? =? X ? =

A. Luca-de, Formal Verification of Probabilistic Systems, 1601.

J. M. Arnaudiès and H. Fraysse, Cours de mathématiques, 4 : Algèbre bilinéaire et géométrie, 1990.

C. Baier, E. M. Clarke, V. Hartonas-garmhausen, and M. Kwiatkowska, Symbolic model checking for probabilistic processes
DOI : 10.1007/3-540-63165-8_199

C. Baier and M. Z. Kwiatkowska, Domain equations for probabilistic processes, Mathematical Structures in Computer Science, vol.10, issue.6, 1997.
DOI : 10.1017/S0960129599002984

A. Bianco and L. De-alfaro, Model checking of probabilistic and nondeterministic systems, FST TCS 95 : Foundations of Software Technology and Theoretical Computer Science, pp.499-513, 1995.
DOI : 10.1007/3-540-60692-0_70

F. Bourdoncle, Sémantiques des Langages Impératifs d'Ordre Supérieur et Interprétation Abstraite, 1992.

C. Baier, M. Kwiatkowska, and G. Norman, Computing probability bounds for linear time formulas over concurrent probabilistic systems, Electronic Notes in Theoretical Computer Science, vol.21, 1999.

J. Chesneaux and J. Vignes, Sur la robustesse de la méthode CESTAC, C. R. Acad. Sci. Paris Sér. I Math, vol.307, issue.16, pp.855-860, 1988.

P. G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation, 1982.

E. M. Clarke, O. Jr, D. A. Grumberg, and . Peled, Model Checking, 1999.

R. Cleaveland, S. A. Smolka, and A. E. Zwarico, Testing preorders for probabilistic processes, Automata, Languages and Programming, 19th International Colloquium, pp.708-719, 1992.
DOI : 10.1007/3-540-55719-9_116

C. Courcoubetis and M. Yannakakis, Markov decision processes and regular events, Proc. ICALP'90, pp.336-349, 1990.
DOI : 10.1007/BFb0032043

P. Cousot, Constructive design of a hierarchy of semantics of a transition system by abstract interpretation, Theoretical Computer Science, vol.277, issue.1-2, 1997.
DOI : 10.1016/S0304-3975(00)00313-3

P. Cousot and R. Cousot, Static determination of dynamic properties of programs, Proceedings of the Second International Symposium on Programming, pp.106-130, 1976.

P. Cousot and R. Cousot, Abstract intrepretation : a unified lattice model for static analysis of programs by construction or approximation of fixpoints, Conference Record of the 4th ACM Symposium on Principles of Programming Languages, pp.238-252, 1977.

P. Cousot, Méthodes itératives de construction et d'approximation de points fixes d'opérateurs monotones sur un treillis, analyse sémantique de programmes, Thèse d'état ès sciences mathématiques, 1978.

P. Cousot and R. Cousot, Constructive versions of Tarski???s fixed point theorems, Pacific Journal of Mathematics, vol.82, issue.1, pp.43-57, 1979.
DOI : 10.2140/pjm.1979.82.43

P. Cousot and R. Cousot, Abstract interpretation and application to logic programs, The Journal of Logic Programming, vol.13, issue.2-3, pp.103-179, 1992.
DOI : 10.1016/0743-1066(92)90030-7

P. Cousot and R. Cousot, Introduction to abstract interpretation Notes de cours du DEA, extraites de, 1998.

P. Cousot and N. Halbwachs, Automatic discovery of linear restraints among variables of a program, Proceedings of the 5th ACM SIGACT-SIGPLAN symposium on Principles of programming languages , POPL '78, 1978.
DOI : 10.1145/512760.512770

D. Dacunha-castelle, Chemins de l'Aléatoire ? Le hasard et le risque dans la société moderne. Champs. Flammarion, 1999.

L. De-alfaro and R. Majumdar, Quantitative solution of omega-regular games, STOC'01, 33rd Annual ACM Symposium on Theory of Computing. Association for Computer Machinery, 2001.
DOI : 10.1016/j.jcss.2003.07.009

M. Luca-de-alfaro, G. Kwiatkowska, D. Norman, R. Parker, and . Segala, Symbolic Model Checking of Probabilistic Processes Using MTBDDs and the Kronecker Representation, TACAS'2000, 2000.
DOI : 10.1007/3-540-46419-0_27

A. Deutsch, Interprocedural may-alias analysis for pointers, Proceedings of the ACM SIGPLAN'94 Conference on Programming Language Design and Implementation (PLDI), pp.230-241, 1994.
DOI : 10.1145/773473.178263

A. Deutsch, Semantic models and abstract interpretation techniques for inductive data structures and pointers, Proceedings of the 1995 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation , PEPM '95, pp.226-229, 1995.
DOI : 10.1145/215465.215594

A. Di, P. , and H. Wiklicky, Concurrent constraint programming : Towards probabilistic abstract interpretation, 2nd International Conference on Principles and Practice of Declarative Programming, 2000.

J. L. Doob, Measure Theory, volume 143 of Graduate Texts in Mathematics, 1994.

P. Granger, Improving the results of static analyses of programs by local decreasing iterations, Foundations of Software Technology and Theoretical Computer Science, 12th Conference, pp.68-79, 1992.
DOI : 10.1007/3-540-56287-7_95

H. Hansson and B. Jonsson, A logic for reasoning about time and reability, 1990.

J. He, K. Seidel, and A. Mciver, Probabilistic models for the guarded command language Formal specifications : foundations, methods, tools and applications, Science of Computer Programming, vol.28, issue.23, pp.171-192, 1995.

W. Hoeffding, Probability Inequalities for Sums of Bounded Random Variables, Journal of the American Statistical Association, vol.1, issue.301, pp.13-30, 1963.
DOI : 10.1214/aoms/1177730491

M. Huth and M. Kwiatkowska, On probabilistic model checking, 1996.

C. Jones, Probabilistic Non-Determinism, 1990.

A. Jung and R. Tix, The Troublesome Probabilistic Powerdomain, Proceedings of the Third Workshop on Computation and Approximation, 1998.
DOI : 10.1016/S1571-0661(05)80216-6

A. S. Kechris, Classical descriptive set theory. Graduate Texts in Mathematics, 1995.

E. Donald and . Knuth, The Art of Computer Programming, 1969.

E. Donald and . Knuth, The Art of Computer Programming, 1969.

D. Kozen, Semantics of probabilistic programs, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979), pp.101-114, 1979.
DOI : 10.1109/SFCS.1979.38

D. Kozen, Semantics of probabilistic programs, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979), pp.328-350, 1981.
DOI : 10.1109/SFCS.1979.38

M. Kwiatkowska and G. Norman, Metric denotational semantics for PEPA, 1996.

M. Kwiatkowska, G. Norman, D. Parker, and R. Segala, Symbolic model checking of concurrent probabilistic systems using mtbdds and simplex, 1999.

M. Z. Kwiatkowska, G. Norman, R. Segala, and J. Sproston, Verifying Quantitative Properties of Continuous Probabilistic Timed Automata, 2000.
DOI : 10.1007/3-540-44618-4_11

M. Z. Kwiatkowska, G. Norman, R. Segala, and J. Sproston, Verifying Quantitative Properties of Continuous Probabilistic Timed Automata, CONCUR 2000 -Concurrency Theory 11th International Conference, number 1877 in Lecture Notes in Computer Science, 2000.
DOI : 10.1007/3-540-44618-4_11

G. Kim, A. Larsen, and . Skou, Bisimulation through probabilistic testing. Information and Computation, pp.1-28, 1991.

L. Michael, A. R. Littman, L. P. Cassandra, and . Kaelbling, Efficient dynamic-programming updates in partially observable markov decision processes, 1995.

G. Lowe, Representing nondeterminism and probabilistic behaviour in reactive processes, 1993.

A. Mciver, Reasoning about efficiency within a probabilistic ??-calculus, Electronic Notes in Theoretical Computer Science, vol.22
DOI : 10.1016/S1571-0661(05)80600-0

A. K. Mciver and C. Morgan, Partial correctness for probabilistic demonic programs, Theoretical Computer Science, vol.266, issue.1-2, 2001.
DOI : 10.1016/S0304-3975(00)00208-5

A. Miné, A New Numerical Abstract Domain Based on Difference-Bound Matrices, Programs as Data Objects, pp.155-172, 2001.
DOI : 10.1007/3-540-44978-7_10

C. John and . Mitchell, Foundations for Programming Languages, 1996.

D. Monniaux, Abstract Interpretation of Probabilistic Semantics, Seventh International Static Analysis Symposium (SAS'00), number 1824 in Lecture Notes in Computer Science, 2000.
DOI : 10.1007/978-3-540-45099-3_17

D. Monniaux, An Abstract Analysis of the Probabilistic Termination of Programs, 8th International Static Analysis Symposium (SAS'01), number 2126 in Lecture Notes in Computer Science, 2001.
DOI : 10.1007/3-540-47764-0_7

D. Monniaux, An abstract Monte-Carlo method for the analysis of probabilistic programs (extended abstract), 28th Symposium on Principles of 236 BIBLIOGRAPHIE Programming Languages (POPL '01), pp.93-101, 2001.

D. Monniaux, Backwards Abstract Interpretation of Probabilistic Programs, European Symposium on Programming Languages and Systems (ESOP '01), number 2028 in Lecture Notes in Computer Science, 2001.
DOI : 10.1007/3-540-45309-1_24

C. Morgan, A. Mciver, K. Seidel, and J. W. Sanders, Probabilistic predicate transformers, ACM Transactions on Programming Languages and Systems, vol.18, issue.3, 1995.
DOI : 10.1145/229542.229547

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.166.127

J. Neveu, Mathematical Foundations of the Calculus of Probabilities, 1965.

J. Neveu, Bases mathématiques du calcul des probabilités, Masson et Cie, 1970.

L. Martin and . Puterman, Markov decision processes : discrete stochastic dynamic programming. Wiley series in probability and mathematical statistics

G. Ramalingam, Data flow frequency analysis, Proceedings of the ACM SIGPLAN '96 Conference on Programming Language Design and Implementation, pp.267-277, 1996.

N. Rasmussen, Reactor safety study ? an assessment of accident risks in u. s. commercial nuclear power plants, Government Printing Office, 1975.

É. Goubault, Static analyses of floating-point operations Lecture Notes in Computer Science, Static Analysis (SAS '01), 2001.

H. Rogers and J. , Theory of recursive functions and effective computability, 1967.

. Rote, Path Problems in Graphs, Computational Graphs Theory, number 7 in Computing supplement, 1990.
DOI : 10.1007/978-3-7091-9076-0_9

R. Y. Rubinstein, Simulation and the Monte-Carlo Method Wiley series in probabilities and statistics, 1981.

W. Rudin, Real and Complex Analysis, 1966.

B. Schneier, Applied Cryptography, 1996.

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, 1996.

R. Segala, Modeling and Verification of Randomized Distributed Real- Time Systems Massachusetts Institute of Technology, 1995.

W. Thomas, Automata on Infinite Objects, pp.135-191, 1990.
DOI : 10.1016/B978-0-444-88074-1.50009-3

P. Thévenod-fosse and H. Waeselynck, Statemate applied to statistical software testing pages 99-109, Proceedings of the 1993 international symposium on Software testing and analysis, pp.99-109, 1993.

J. Vignes, A stochastic arithmetic for reliable scientific computation, Mathematics and Computers in Simulation, vol.35, issue.3, pp.233-261, 1993.
DOI : 10.1016/0378-4754(93)90003-D

J. Vignes and R. Alt, An efficient stochastic method for round-off error analysis, Accurate scientific computations, pp.183-205, 1985.
DOI : 10.1007/3-540-16798-6_12

G. Winskel, The Formal Semantics of Programming Languages, 1993.

A. Abstract and T. , 120 9.3 Experimental results Experimental results with bigger output samples, p.126

.. Abstract-addition-of-sub-exponentials, 158 12.1 An extended Gaussian distribution 164 12.2 No greatest lower bound among parabolas 168 12.5 Common upper bounds for two ellipses 174 13.1 A simple probabilistic program 187 13.3 Another probabilistic program, 157 11.2 Abstract addition of sub 165 12.3 Common lower bounds in quadratic polynomials . . . . . . . . . 167 12.4 Common upper bound in Gaussians 189 14.1 Computing ?/4 using the Monte-Carlo method. . . . . . . . . . . 198 14.2 Computing ?/4 using an approximate Monte-Carlo method. . . . 202 14.3 Upper bound on the probability that the probability estimate exceeds the real value by more than t, p.208