## CONCENTRATION INEQUALITIES AND ESTIMATION OF CONDITIONAL PROBABILITIES

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@MISC{Maume-deschamps_concentrationinequalities,

author = {V. Maume-deschamps},

title = {CONCENTRATION INEQUALITIES AND ESTIMATION OF CONDITIONAL PROBABILITIES},

year = {}

}

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### Abstract

Abstract. We prove concentration inequalities inspired from [DP] to obtain estimators of conditional probabilities for weak dependant sequences. This generalize results from Csiszár ([Cs]). For Gibbs measures and dynamical systems, these results lead to construct estimators of the potential function and also to test the nullity of the asymptotic variance of the system. This paper deals with the problems of typicality and conditional typicality of “empirical probabilities ” for stochastic process and the estimation of potential functions for Gibbs measures and dynamical systems. The questions of typicality have been studied in [FKT] for independent sequences, in [BRY, R] for Markov chains. In order to prove the consistency of estimators of transition probability for Markov chains of unknown order, results on typicality and conditional typicality for some (Ψ)-mixing process where obtained in [CsS, Cs]. Unfortunately, lots of natural mixing process do not satisfy this Ψ-mixing condition (see [DP]). We consider a class of mixing process inspired from [DP]. For this class, we prove strong typicality and strong conditional typicality. In the particular case of Gibbs measures (or complete connexions chains) and for certain dynamical systems, from the typicality results we derive an estimation of the potential as well as procedure to test the nullity of the asymptotic variance of the process. More formally, we consider X0,...., Xn,... a stochastic process taking values on an complete set Σ and a sequence of countable partitions of Σ, (Pk)k∈N such that if P ∈ Pk then there exists a unique � P ∈ Pk−1 such that almost surely, Xj ∈ P ⇒ Xj−1 ∈ � P. Our aim is to obtain empirical estimates on the probabilities: P(Xj ∈ P), P ∈ Pk, and the conditional probabilities:

### Citations

314 | The minimum description length principle in coding and modeling - Barron, Rissanen, et al. - 1998 |

57 | The consistency of the BIC Markov order estimator - Csisz'ar, Shields - 2000 |

48 |
New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132
- Dedecker, Prieur
- 2005
(Show Context)
Citation Context ...er, results on typicality and conditional typicality for some (Ψ)-mixing process where obtained in [CsS, Cs]. Unfortunately, lots of natural mixing process do not satisfy this Ψ-mixing condition (see =-=[DP]-=-). We consider a class of mixing process inspired from [DP]. For this class, we prove strong typicality and strong conditional typicality. In the particular case of Gibbs measures (or complete connexi... |

35 | Decay of correlations for nonHölderian dynamics. A coupling approach
- Bressaud, Fernández, et al.
- 1999
(Show Context)
Citation Context ... (a, x0, . . .) ∈ Σ if and only if ay ∈ Σ for all y ∈ Ax0 . For x ∈ Σ, let g(x) = P(X0 = x0|Xi = xi, i ≥ 1). We shall prove that ˆgn,k is a consistent estimator of g. This is known (see [KMS], [Ma2], =-=[BGF]-=-, [Po]) that such a process is mixing for suitable functions. Let γ⋆ � n = k≥n γk, define a distance on Σ by d(x, y) = γ⋆ n if and only if xj = yj for j = 0, . . . , n − 1 and xn �= yn. Let L be the s... |

34 |
Transformations dilatantes de l’intervalle et théorèmes limites
- Broise
- 1996
(Show Context)
Citation Context ...ssumptions, these results on existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing remain valid, with an infinite countable partition (=-=[Br]-=-, [L,S,V], [Ma1]). Theorem 2.3. ([Sc], [Co], [Li]) Let C be the space of functions on [0, 1] of bounded variations. Let T satisfy the assumptions 2. Then we have the following mixing property : there ... |

32 |
Universal properties of maps on an interval
- Collet, Eckmann, et al.
- 1986
(Show Context)
Citation Context ...onditions are sufficient to ensure existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing (see for example [Sc] for the Markov case and =-=[Co]-=-, [Li] for the general case). Under more technical assumptions, these results on existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing ... |

30 | Dynamical sources in information theory: fundamental intervals and word prefixes, Algorithmica 29
- Vallée
- 2001
(Show Context)
Citation Context ... the process is zero. This is motivated by the fact that for the process studied in the previous sections, we may prove a central limit theorem provided the asymptotic variance is not zero (see [Br], =-=[V]-=- for examples). We are concerned with a process (Xj)j∈N satisfying Conditions of Section 2.1 or Section 3. We assume moreover that the system is complete that is : T (Ai) = I for all i if we are in th... |

25 |
A new covariance inequality and applications. Stochastic Process
- Dedecker, Doukhan
(Show Context)
Citation Context ...o Lemma 4 in [DP]. Lemma 1.2. ΦC(k) = sup {�E(ϕ(Xi+k)|Mi) − E(ϕ(Xi+k))�∞ , ϕ ∈ C1} . We postpone the proof of Lemma 1.2 to the end of the proof of the proposition. Secondly, we apply Proposition 4 in =-=[DD]-=- to get : (let Yi = ϕ(Xi)−E(ϕ(Xi))) �Sn(ϕ) − E(Sn(ϕ))�p ≤ ≤ ≤ � � � 2p 2p 2p n� max i≤ℓ≤n �Yi i=1 n� i=1 n� i=1 �Yi� p 2 ℓ� k=i . � . E(Yk|Mi)� p 2 n� �E(Yk|Mi)�∞ (we have used that by Lemma 1.2, �E(Y... |

24 |
Large-scale typicality of Markov sample paths and consistency of MDL order estimators
- Csiszár
- 2002
(Show Context)
Citation Context ... MAUME-DESCHAMPS Abstract. We prove concentration inequalities inspired from [DP] to obtain estimators of conditional probabilities for weak dependant sequences. This generalize results from Csiszár (=-=[Cs]-=-). For Gibbs measures and dynamical systems, these results lead to construct estimators of the potential function and also to test the nullity of the asymptotic variance of the system. This paper deal... |

23 | Vaienti Conformal measure and decay of correlations for covering weighted systems Ergodic th. & dynam. syst - Liverani, Saussol, et al. - 1998 |

22 | Decay of correlations for piecewise expanding maps
- Liverani
- 1995
(Show Context)
Citation Context ...ons are sufficient to ensure existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing (see for example [Sc] for the Markov case and [Co], =-=[Li]-=- for the general case). Under more technical assumptions, these results on existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing remain... |

21 |
Exponential inequalities for dynamical measures of expanding maps of the interval, Probab
- Collet, Mart́ınez, et al.
- 2002
(Show Context)
Citation Context ... µ(D k(n) ∪ E k(n)) + 2 p µ(|ˆgn(x) − g(x)| > t) + t p ≤ Ct2 p � γ κ ≤ Ct2 p � γ κ � k(n) � k(n) + 2 p (4e −Lt2 n 1−ε + 2 p (4e −Ln1−ε−2α + 2e −Ln1−ε ) + t p + 2e −Ln1−ε ) + Ctn −pα ln ℓ γ Remark. In =-=[CMS]-=-, an exponential inequality is proven for Lipschitz functions of several variables for expanding dynamical systems of the interval. We can not use such a result here because characteristic functions o... |

14 | Deviations from uniformity in random strings. Probability Theory and Related Fields - Flajolet, Kirschenhofer, et al. - 1988 |

9 |
2000), Iterates of expanding maps
- Barbour, Gerrard, et al.
(Show Context)
Citation Context ...· · , Y0) as the same law as (X0, · · · , Xn), then Cov(ψ(Xi), ϕ(Xn + i)) = Cov(ψ(Yi+n), ϕ(Yi)) and the process (Yn)n∈N satisfies our Assumptions 1. Using the stationarity, it satisfies also(**), see =-=[BGR]-=- and [DP] for more developments on this “trick”. Applying Theorem 1.5 to the process (Yn)n∈N and using that n� j=1 1P (Yj) Law = n−1 � 0 1P (Xj)s8 V. MAUME-DESCHAMPS and n−2 � j=0 we obtain the follow... |

9 | Correlation decay for Markov maps on a countable state space
- MAUME-DESCHAMPS
(Show Context)
Citation Context ...se results on existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing remain valid, with an infinite countable partition ([Br], [L,S,V], =-=[Ma1]-=-). Theorem 2.3. ([Sc], [Co], [Li]) Let C be the space of functions on [0, 1] of bounded variations. Let T satisfy the assumptions 2. Then we have the following mixing property : there exists C > 0, 0 ... |

3 |
Propriétés Statistiques de Systèmes Dynamiques Non Markoviens
- Paccaut
- 2000
(Show Context)
Citation Context ...Assume that Cmax = max C(1Aj ) < ∞. There exists 0 < j=1,..., γ < 1 such that for all P ∈ Pk, we have µ(P ) ≤ γ k . Proof. The proof of Lemma 2.2 follows from the mixing property. It is inspired from =-=[Pa]-=-. Let n0 ∈ N to be fixed later. Let P ∈ Pk, for some indices i0, . . . , ik−1, we have that Then, let ℓ = [ k n0 ], P = {x ∈ Ai0 , . . . , T k−1 x ∈ Aik−1 }. µ(P ) = P(X0 ∈ Ai0 , . . . , Xk−1 ∈ Aik−1 ... |

2 |
Vitesse de convergence vers l’état d’équlibre pour des dynamiques markoviennes non höldériennes
- KONDAH, MAUME, et al.
- 1997
(Show Context)
Citation Context ... , ) ∈ Σ, ax = (a, x0, . . .) ∈ Σ if and only if ay ∈ Σ for all y ∈ Ax0 . For x ∈ Σ, let g(x) = P(X0 = x0|Xi = xi, i ≥ 1). We shall prove that ˆgn,k is a consistent estimator of g. This is known (see =-=[KMS]-=-, [Ma2], [BGF], [Po]) that such a process is mixing for suitable functions. Let γ⋆ � n = k≥n γk, define a distance on Σ by d(x, y) = γ⋆ n if and only if xj = yj for j = 0, . . . , n − 1 and xn �= yn. ... |

2 |
Ergodic theory and thermodynamic of one-dimensional Markov expanding endomorphisms. Dynamical systems
- Schmitt
- 1996
(Show Context)
Citation Context ... N(k) P = [0, 1]. The above conditions are sufficient to ensure existence and uniqueness of an absolutely continuous invariant measure as well as an estimation of the speed of mixing (see for example =-=[Sc]-=- for the Markov case and [Co], [Li] for the general case). Under more technical assumptions, these results on existence and uniqueness of an absolutely continuous invariant measure as well as an estim... |

1 | Statistical properties of Markov dynamical sources: applications to information theory
- Chazal, Maume-Deschamps
(Show Context)
Citation Context ...s well as σ : Σ → Σ. Definition 2. ([Br]) Let n−1 � Sn = (Xj − E(X0)) and Mn = j=0 � � �2 Sn √n dP. The sequence Mn converges to V which we shall call the asymptotic variance. Proposition 4.1. ([Br], =-=[CM]-=-) The asymptotic variance V is zero if and only if the potential log g is a cohomologue to a constant : log g = log a + u − u ◦ T , with a > 0, u ∈ BV or u ∈ L. Because we are in a stationary setting,... |

1 |
Propriétés de mélange pour des sytèmes dynamiques markoviens. Thèse de l’Université de Bourgogne, available at http://math.u-bourgogne.fr/IMB/maume
- Maume-Deschamps
(Show Context)
Citation Context ...Σ, ax = (a, x0, . . .) ∈ Σ if and only if ay ∈ Σ for all y ∈ Ax0 . For x ∈ Σ, let g(x) = P(X0 = x0|Xi = xi, i ≥ 1). We shall prove that ˆgn,k is a consistent estimator of g. This is known (see [KMS], =-=[Ma2]-=-, [BGF], [Po]) that such a process is mixing for suitable functions. Let γ⋆ � n = k≥n γk, define a distance on Σ by d(x, y) = γ⋆ n if and only if xj = yj for j = 0, . . . , n − 1 and xn �= yn. Let L b... |

1 |
Rates of mixing for potentials of summable variation
- Policott
(Show Context)
Citation Context ..., . . .) ∈ Σ if and only if ay ∈ Σ for all y ∈ Ax0 . For x ∈ Σ, let g(x) = P(X0 = x0|Xi = xi, i ≥ 1). We shall prove that ˆgn,k is a consistent estimator of g. This is known (see [KMS], [Ma2], [BGF], =-=[Po]-=-) that such a process is mixing for suitable functions. Let γ⋆ � n = k≥n γk, define a distance on Σ by d(x, y) = γ⋆ n if and only if xj = yj for j = 0, . . . , n − 1 and xn �= yn. Let L be the space o... |

1 | non k-Markov points.Stochastic complexity in statistical inquiry - Rissanen - 1989 |