Propriétés asymptotiques des processus de branchement en environnement aléatoire

Let (Z_n) be a branching process in a random environment. If the process in sub-critical or critical, we study the convergence rate of the survival probability P(Z_n>0) as n→∞; if the process is supercritical, we give a necessary and sufficient condition for the convergence in L^p of the natural martingale, where p>1 is given.     

The asymptotic ruin problem when the healthy and sick periods form an alternating renewal process

Let {T₁, Y₁}^∞_i=1 be a sequence of positive independent random variables. Let, also, Z₁ = βY′₁ ? πT_i, i = 1, 2, …, where Y′₁ = Max(0, Y_i ? w), w ? 0, and where β < 0 and π is such that E(Z₁) < 0. We consider the random walk of partial sums S_n = ?ⁿ_i=1Z_i in the presence of an absorbing region (u, ∞), u ? 0, and S₀ ≡ 0. Of interest is $\text{ψ(u) = Pr(}\text{S}\text{?}\text{≤ u)}$ where S? = Sup(0, S₁, S₂, …, S_n, …).     

Existence of triple solutions of discrete (n,p) boundary value problems

We consider the following boundary value problem: −Δⁿy = F(k,y, Δy,…,Δⁿ⁻¹y), k ϵ Z[n − 1, N], Δⁱy(0) = 0, 0 ≤ i ≤ n − 2, Δ^py(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.     

On the moments of some first passage times for sums of dependent random variables

Let S_n,n = 1, 2, …, denote the partial sums of integrable random variables. No assumptions about independence are made. Conditions for the finiteness of the moments of the first passage times N(c) = min {n: S_n>ca(n)}, where c ≥ 0and a(y) is a positive continuous function on [0, ∞), such that a(y) = o(y)as y → ∞, are given. With the further assumption that a(y) = yP,0 ≤ p < 1, a law of large numbers and the asymptotic behaviour of the moments when c → ∞ are obtained. The corresponding stopped sums are also studied.     

Finite Groups Whose <Emphasis Type="Italic">n</Emphasis>-Maximal Subgroups Are Modular

Let G be a finite group. If M_n< M_n?1< · · · < M₁< M₀ = G with Mi a maximal subgroup of M_i?1 for all i = 1,..., n, then M_n (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.     

Large deviations for sums indexed by the generations of a Galton–Watson process

In this paper, we study the large deviation behavior of sums of i.i.d. random variables X _i , where Z _n is the nth generation of a supercritical Galton–Watson process. We assume the finiteness of the moments and EZ ₁ logZ ₁ . The underlying interplay of large deviation probabilities of partial sums of the X _i and of lower deviation probabilities of Z is clarified. Here, we heavily use lower deviation probability results on Z we recently published in [7]. This paper has been written during the time the second author was a staff member of the WIAS Berlin.     

Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

As is well known, for a supercritical Galton-Watson process Z _n whose offspring distribution has mean m > 1, the ratio W _n := Z _n /m ⁿ has almost surely a limit, say W. We study the tail behaviour of the distributions of W _n and W in the case where Z ₁ has a heavy-tailed distribution, that is, $\mathbb{E}e^{\lambda {\rm Z}_1 } = \infty $ for every λ > 0. We show how different types of distributions of Z ₁ lead to different asymptotic behaviour of the tail of W _n and W. We describe the most likely way in which large values of the process occur.     

A local limit theorem for the number of nodes,the height,and the number of final leaves in a critical branching process tree

Let Z_i be the number of particles in the ith generation of a non-degenerate critical Bienaymé-Galton-Watson process with offspring distribution $ p_r = P \{\hbox{a given individual has {\it r} children}\},\kern2em r\geq 0. $ Let ν = Σ^infinity₀ Z_j be the total progeny and let ζ = inf{r: Z_r = 0} be the extinction time. Equivalently, ν and ζ are the total number of nodes and (1 + the height), respectively, of the family tree of the branching process. Assume that E{Z₁} = Σ p_rr = 1 and E{Z₁^{3 + δ}} = Σ p_rr^{3 + δ} < infinity for some δ ϵ (0, 1). We find an asymptotic formula with remainder term for k⁴P{ζ = k + 1, Z_k = ℓ ν = n} when k→ infinity, which is uniform over n and ℓ. This is used to confirm a conjecture by Wilf that the number of leaves in the last generation of a randomly chosen rooted tree converges in distribution. More precisely, in the terminology introduced above, there exists a probability distribution {q₁} such that for n → infinity $ P\{Z_{\zeta-1} = l | \nu=n\} = q_l + O \left({{\log^3 n } \over {n^{1/2}}}\right), $ uniformly over ℓ ≥1. The limiting distribution is identified by means of a functional equation for the generating function Σ^infinity₁ q s^ℓ. Numerically, q₁ ≅ 0.0602, q₂ ≅ 0.248, q₃ ≅ 0.094, and q₄ ≅ 0.035. Our method can also be used to find lim _{k→ infinity} k⁴P{ζ = k + 1, Z_k = ℓ ν = n} when only E{Z₁^{2 + δ}} < infinity for some 0 ≤δ≤1, but we do not treat this case here; it goes without saying that the fewer moment assumptions one makes, the poorer the estimates become. © 1996 John Wiley & Sons, Inc.     

Normalisation d'un processus de branchement critique dans un environnement aléatoire

Let (Z_n) be a critical branching process in an independent and identically distributed (i.i.d.) random environment. For each fixed environment ω, let C_n=E^ω[Z_n∣Z_n>0] be the conditional expectation of Z_n given Z_n>0. We prove an analogue of Yaglom's law: as n→∞, the conditional law of Z_n/C_n, conditional on Z_n>0, converges to a non-degenerate law on [0,∞). We give also an analogue of Kolmogorov's law, as well as a local limit theorem for the semi-group of probability generating functions. To cite this article: Y. Guivarc'h et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

How do random Fibonacci sequences grow?

We study the random Fibonacci sequences defined by ${F_1 = F_2 = \widetilde F_1 = \widetilde F_2 = 1}$ and for n ≥ 1, F _n+2 = F _n+1 ± F _n (linear case) and ${\widetilde F_{n+2} = |\widetilde F_{n+1} \pm \widetilde F_{n}|}$ (non-linear case), where each ± sign is independent and either + with probability p or ? with probability 1 ? p (0 < p ≤ 1). Our main result is that the exponential growth of F _n for 0 < p ≤ 1, and of ${\widetilde F_{n}}$ for 1/3 ≤ p ≤ 1 is almost surely given by $${\int \limits _0^\infty \log x\, d\nu_\alpha (x),}$$ where α is an explicit function of p depending on the case we consider, and ν _α is an explicit probability distribution on ${\mathbb{R}_+}$ defined inductively on Stern–Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of p, since we prove that it is equal to zero for 0 < p ≤ 1/3. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.     

Periodic regeneration

We consider stochastic processes Z = (Z_t)_[0,∞), on a general state space, having a certain periodic regeneration property: there is an increasing sequence of random times (S_n)^∞₀ such that the post-S_n process is conditionally independent of S₀,…,S_n given (S_n mod 1) and the conditional distribution does not depend on n. Our basic condition is that the distributions $\text{P}{}_{\mathrm{s}}\text{(A) =}\text{P}\text{(S}{}_{\mathrm{n+1}}\text{? S}{}_{\mathrm{n}}\text{∈ A|S}{}_{\mathrm{n}}\text{= s), s ∈[0, 1)}$, have a common component that is absolutely continuous w.r.t. Lebesgue measure. Then Z has the following time-homogeneous regeneration property: there exists a discrete aperiodic renewal process T = (T_n)^∞₀ such that the post-T_n process is independent of T₀,…,T_n and its distribution does not depend on n; this yields weak ergodicity. Further, the Markov chain (S_nmod 1)^∞₀ has an invariant distribution π_[0,1) and it holds that T_n+1 ? T_n has finite first moment if and only if m = ∫ m(P_s)π_[0,1)(ds) < ∞ where m(P_s) is the first moment of P_s; this yields periodic ergodicity. Also, some distributional properties of T₀ and T_n+1 ? T_n are established leading to improved ergodic regults. Finally, a uniform key periodic renewal theorem is derived.     

Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes

Let X ₁,..., X _n, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H ₀: X ₁,..., X _n is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H ₀ versus the “at most one change (AMOC) in the mean” alternative hypothesis H _A: there is an integer k*, 1 ≤ k* < n, such that EX ₁ = · · · = EX_k* ≠ EX_k*+1 = ··· = EX _n. A natural way of testing for H ₀ versus H _A is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δ_k = |S _k/k - (S _n - S _k)/(n - k)|, where S _k:= X ₁ + ··· + X _k. Asymptotic distributions of these test statistics under H ₀ as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Z _n(t)|, where {Z _n(t), 0 ≤ t ≤ 1}_n≥1 are the tied-down partial sums processes such that

$${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n $$

if 0 ≤ t < 1, and Z _n(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H ₀ versus H _A via sup-functionals of appropriately weighted |Z _n(t)|. Simply considering max_1?k<n Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Z _n(t)|/(t(1 - t)) under H ₀ and assuming also that E|X|^r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Z _n(t)| has not been considered before.

     

A generalization of the Chung-Erdös inequality for the probability of a union of events

A generalization of the Chung-Erdös inequality for the probability of a union of arbitrary events is proved using some lower bounds for tail probabilities. We present a lower bound for the probability of appearance of at least m events from a set of events A₁,..., A_n, where 1 ≤ m ≤ n. Bibliography: 6 titles.     

Bounded branching process and and/or tree evaluation

We study the tail distribution of supercritical branching processes for which the number of offspring of an element is bounded. Given a supercritical branching process {Z_n} with a bounded offspring distribution, we derive a tight bound, decaying super-exponentially fast as c increases, on the probability Pr[Z_n > cE(Z_n)], and a similar bound on the probability Pr[Z_n ≤ E(Z_n)/c] under the assumption that each element generates at least two offspring. As an application, we observe that the execution of a canonical algorithm for evaluating uniform AND/OR trees in certain probabilistic models can be viewed as a two-type supercritical branching process with bounded offspring, and show that the execution time of this algorithm is likely to concentrate around its expectation, with a standard deviation of the same order as the expectation.     

Weighted moments of the limit of a branching process in a random environment

Let (Z _n ) be a supercritical branching process in an independent and identically distributed random environment ζ = (ζ ₀, ζ ₁,…), and let W be the limit of the normalized population size Z _n / $\mathbb{E}$ (Z _n |ζ). We show a necessary and sufficient condition for the existence of weighted moments of W of the form $\mathbb{E}$ , where α ≥ 1 and ? is a positive function slowly varying at ∞.     

The oscillating random walk

{Y_n;n=0, 1, …} denotes a stationary Markov chain taking values in R^d. As long as the process stays on the same side of a fixed hyperplane E⁰, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′_n = |Y′_n?1?Z_n| (with the Z_n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y_n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.     

Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

For a blockwise martingale difference sequence of random elements {Vn , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞∑ n i=1 Vi /gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.     

On a class of martingale inequalities

Let Z = {Z₀, Z₁, Z₂,…} be a martingale, with difference sequence X₀ = Z₀, X_i = Z_i ? Z_{i ? 1}, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(sup_i |X_i| ≥ λ) ≤ C sup_iE |Z_i|, for λ > 0, is C = (log 2)^?1. If Z is finite of length n, it is proved that the best constant is $\text{C}{}_{\mathrm{n}}\text{= [n(2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1)]}{}^{\mathrm{?1}}$. The analogous best constant C_n(z) when Z₀ ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity G_n(z, E) = sup_zP(max_i=1,…,n |X_i| ≥ 1), the supremum being taken over all martingales Z with Z₀ ≡ z and E|Z_n| = E. The expression for G_n(z,E) is derived by induction, using methods from the theory of moments.     

Haar-Based Multiresolution Stochastic Processes

Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = \sum_{n=0}^\infty\lambda_n\varDelta _n(t)\epsilon_n,$$ where ?? _n ?? _n (t) is the integral of the nth Haar wavelet from 0 to t, and ?? _n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for X _t to converge uniformly with probability one. Each stochastic process , the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of , the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as H?lder continuity and fractional dimensions of graphs of the processes.     

Extinction conditions for certain bisexual Galton-Watson branching processes

Summary Consider a two-type Galton-Watson branching process with X _nfemales and Y _nmales in the nth generation, modified so that the (X inn}+1, Y n}+1) offspring in the (n + l)th generation are derived from Z _n=(X _n, T_n) mating units which reproduce independently with the same offspring distribution for every mating unit in every generation. Necessary and sufficient conditions are found for the process to become extinct (i. e., Z _N= 0 for some positive integer N) with probability one for the two particular mating functions (x, y) = x min(1, y), which corresponds to perfect promiscuity, and (x, y) = min(x, dy) (d a positive integer), which corresponds to polygamous mating with d wives per husband when there is an abundance of females. Essentially, the condition for pr(Z _n 0)=1 to be true is the commonsense condition that E(Z ₁¦Z₀=j) j for all sufficiently large j.