Bifurcating Markov Chains Research Articles

Subexponential Concentration Inequalities for Bifurcating Markov Chains

In this paper, for bifurcating Markov chains, under weak and general conditions, we establish explicit subexponential concentration inequalities for different types of empirical means of unbounded functions. The subexponential concentration captures the tail behavior of distributions that are less concentrated than the Gaussian distribution. In particular cases, our results recover the existing subgaussian concentration.

Central limit theorem for bifurcating Markov chains: the mother-daughters triangles case

The main objective of this article is to establish a central limit theorem for additive three-variable functionals of bifurcating Markov chains. We thus extend the central limit theorem under point-wise ergodic conditions studied in S.V. Bitseki Penda and J.-F. Delmas [Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions, Ann. Appl. Probab. 32(5) (2022), pp. 3817–3849] and to a lesser extent, the results of S.V. Bitseki Penda and J.-F. Delmas [Central limit theorem for bifurcating Markov chains under L 2 -ergodic conditions, Adv. Appl. Probab. 54(4) (2022), pp. 999–1031] on central limit theorem under L 2 ergodic conditions. Our results also extend and complement those of J. Guyon [Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, Ann. Appl. Probab. 17(5–6) (2007), pp. 1538–1569] and J.-F. Delmas and L. Marsalle [Detection of cellular aging in a Galton-Watson process, Stochastic Process. Appl. 120(12) (2010), pp. 2495–2519]. In particular, when the ergodic rate of convergence is greater than 1 / 2 , we have, for certain class of functions, that the asymptotic variance is non-zero at a speed faster than the usual central limit theorem studied by Guyon and Delmas-Marsalle.

Kernel estimation of the transition density in bifurcating Markov chains

We study the kernel estimators of the transition density of bifurcating Markov chains. Under some ergodic and regularity properties, we prove that these estimators are consistent and asymptotically normal. Next, in the numerical studies, we propose two data-driven methods to choose the bandwidth parameters. These methods, based on the so-called two bandwidths approach, are adaptation for bifurcating Markov chains of the least squares Cross-Validation and the rule of thumb method. Finally, we provide an example with real data.

The asymptotic equipartition property for a special Markov random field

The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we study the generalized asymptotic equipartition property in the form of moving average for N bifurcating Markov chains indexed by an N-branch Cayley tree, which is a special case of Markov Urandom fields. Firstly, we construct a class of random variables containing a parameter with means of 1, and establish a strong limit theorem for the moving average of multivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we present the strong law of large numbers for the frequencies of occurrence of states of the moving average, as well as the generalized asymptotic equipartition property for N bifurcating Markov chains indexed by an N-branch Cayley tree. As corollaries, we also generalize some known results.

Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains

Bitseki and Delmas (2022) have studied recently the central limit theorem for kernel estimator of invariant density in bifurcating Markov chains. We complete their work by proving a moderate deviation principle for this estimator. Unlike the work of Bitseki and Gorgui (2022), it is interesting to see that the distinction of the two regimes disappears and that we are able to get moderate deviation principle for large values of the ergodic rate. It is also interesting and surprising to see that for moderate deviation principle, the ergodic rate begins to have an impact on the choice of the bandwidth for values smaller than in the context of central limit theorem studied by Bitseki and Delmas (2022).

Central Limit Theorem for Kernel Estimator of Invariant Density in Bifurcating Markov Chains Models

Bifurcating Markov chains are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistency and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations of average over large generations) disappears. This result is a first step to go beyond the threshold condition on the ergodic rate given in previous statistical papers on functional estimation.

Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (Ann. Appl. Probab. 17 (2007) 1538–1569), where the considered additive functionals are sums of martingale increments, and only one regime appears. Our result can be seen as a discrete time version, but with general trait evolution, of results in the time continuous setting of branching particle system from Adamczak and Miłoś (Electron. J. Probab. 20 (2015) 42), where the evolution of the trait is given by an Ornstein–Uhlenbeck process.

Moderate deviation principles for bifurcating Markov chains: case of functions dependent of one variable

The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout & Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki & Delmas (2020) about the central limit theorem for general additive functionals of bifurcating Markov chains, we give here a moderate deviation principle for additive functionals of bifurcating Markov chains when the functions depend on one variable. This work is done under the uniform geometric ergodicity and the uniform ergodic property based on the second spectral gap assumptions. The proofs of our results are based on martingale decomposition recently developed by Bitseki & Delmas (2020) and on results of Dembo (1996), Djellout (2001) and Puhalski (1997).

The Generalized Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree.

In this paper, we study the generalized entropy ergodic theorem for nonhomogeneous bifurcating Markov chains indexed by a binary tree. Firstly, by constructing a class of random variables with a parameter and the mean value of one, we establish a strong limit theorem for delayed sums of the bivariate functions of such chains using the Borel–Cantelli lemma. Secondly, we prove the strong law of large numbers for the frequencies of occurrence of states of delayed sums and the generalized entropy ergodic theorem. As corollaries, we generalize some known results.

Equivalent properties for the bifurcating Markov chains indexed by a binary tree

Guyon has introduced an important model of bifurcating Markov chains indexed by a binary tree taking values in general state space and studied their limit theorems, and applied these results to detect cellular aging. In this paper, we will establish some equivalent properties and a property for this Markov chains. From these properties one can find that two daughters only depend on their mother for bifurcating Markov chains indexed by a binary tree.

A class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree

In this paper, we first introduce the asymptotic logarithmic likelihood ratio as a measure of the deviation between the arbitrary random fields and the bifurcating Markov chain on a binary tree. Then a class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree is established by constructing a nonnegative martingale. As corollaries, we obtain the strong law of large numbers (SLLN) and the asymptotic equipartition property (AEP) for the bifurcating Markov chains indexed by a binary tree.

The strong law of large numbers for non homogeneous M-bifurcating Markov chains indexed by a M-branch Cayley tree

ABSTRACTThis article is devoted to the strong law of large numbers and the entropy ergodic theorem for non homogeneous M-bifurcating Markov chains indexed by a M-branch Cayley tree, which generalizes the relevant results of tree-indexed nonhomogeneous bifurcating Markov chains. Meanwhile, our proof is quite different from the traditional method.

Transportation and concentration inequalities for bifurcating Markov chains

We investigate the transportation inequality for bifurcating Markov chains which are a class of processes indexed by a regular binary tree. Fitting well models like cell growth when each individual gives birth to exactly two offsprings, we use transportation inequalities to provide useful concentration inequalities. We also study deviation inequalities for the empirical means under relaxed assumptions on the Wasserstein contraction for the Markov kernels. Applications to bifurcating nonlinear autoregressive processes are considered for point-wise estimates of the non-linear autoregressive function.

Autoregressive functions estimation in nonlinear bifurcating autoregressive models

Bifurcating autoregressive processes, which can be seen as an adaptation of au-toregressive processes for a binary tree structure, have been extensively studied during the last decade in a parametric context. In this work we do not specify any a priori form for the two autoregressive functions and we use nonparametric techniques. We investigate both nonasymp-totic and asymptotic behavior of the Nadaraya-Watson type estimators of the autoregressive functions. We build our estimators observing the process on a finite subtree denoted by Tn, up to the depth n. Estimators achieve the classical rate |Tn| −β/(2β+1) in quadratic loss over Holder classes of smoothness. We prove almost sure convergence, asymptotic normality giving the bias expression when choosing the optimal bandwidth and a moderate deviations principle. Our proofs rely on specific techniques used to study bifurcating Markov chains. Finally, we address the question of asymmetry and develop an asymptotic test for the equality of the two autoregressive functions.

Adaptive estimation for bifurcating Markov chains

In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under $L^p$ -loss error, for $1\leq p<\infty$. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

The Strong Law of Large Numbers and the Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree

Guyon (Guyon J. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann Appl Probab, 2007, 17: 1538-1569) introduced an important model for homogeneous bifurcating Markov chains indexed by a binary tree taking values in general state space and studied their limit theorems. The results were applied to detect cellular aging. In this paper, we define a discrete form of nonhomogeneous bifurcating Markov chains indexed by a binary tree and discuss the equivalent properties for them. The strong law of large numbers and the entropy ergodic theorem are studied for these Markov chains with finite state space. In contrast to previous work, we use a new approach to prove the main results of this paper.

Deviation inequalities for bifurcating Markov chains on Galton−Watson tree

We are interested in bifurcating Markov chains on Galton−Watson tree. These processes are an extension of bifurcating Markov chains, which was introduced by Guyon to detect cellular aging from cell lineage, in case the index set is a binary Galton−Watson process. First, under geometric ergodicity assumption of an embedded Markov chain, we provide polynomial deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton−Watson tree. Next, under some uniformity, we derive exponential inequalities. These results allow to exhibit different regimes of convergence which correspond to a competition between the geometric ergodic speed of the underlying Markov chain and the exponential growth of the Galton−Watson tree. As application, we derive deviation inequalities (for either the Gaussian setting or the bounded setting) for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data which allow, in the case of cell division, to take into account the cell’s death.

Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application

First, under a geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for the bifurcating Markov chains (BMC). The BMC model was introduced by Guyon to detect cellular aging from cell lineage, and our aim is thus to complete his asymptotic results. The deviation inequalities are then applied to derive first result on the moderate deviation principle (MDP) for a functional of the BMC with a restricted range of speed, but with a function which can be unbounded. Next, under a uniform geometric ergodicity assumption, we provide deviation inequalities for the BMC and apply them to derive a second result on the MDP for a bounded functional of the BMC with a larger range of speed. As statistical applications, we provide superexponential convergence in probability and deviation inequalities (for either the Gaussian setting or the bounded setting), and the MDP for least square estimators of the parameters of a first-order bifurcating autoregressive process.