Martin-Löf randomness and Galton–Watson processes

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] is robust, but surprisingly, that the effective continued fraction dimension and effective (base- b ) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that effective dimension is invariant with respect to base conversion [ 8 ]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\) but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the “non-faithfulness” of certain families of covers [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.

The members of Martin-Lof random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Lof random Galton-Watson trees with survival parameter $\frac{2}{3}$. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than $\gamma=\log_2 \frac{3}{2}$, and a necessary condition is to have effective Hausdorff dimension greater than or equal to *** .

Controlled Branching Processes

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.

If viruses or other pathogens infect a single host, the outcome of infection often hinges on the fate of the initial invaders. The initial basic reproduction number R0, the expected number of cells infected by a single infected cell, helps determine whether the initial viruses can establish a successful beachhead. To determine R0, the Kingman coalescent or continuous-time birth-and-death process can be used to infer the rate of exponential growth in an historical population. Given M sequences sampled in the present, the two models can make the inference from the site frequency spectrum (SFS), the count of mutations that appear in exactly k sequences (k=1,2,…,M). In the case of viruses, however, if R0 is large and an infected cell bursts while propagating virus, the two models are suspect, because they are Markovian with only binary branching. Accordingly, this article develops an approximation for the SFS of a discrete-time branching process with synchronous generations (i.e., a Galton–Watson process). When evaluated in simulations with an asynchronous, non-Markovian model (a Bellman–Harris process) with parameters intended to mimic the bursting viral reproduction of HIV, the approximation proved superior to approximations derived from the Kingman coalescent or continuous-time birth-and-death process. This article demonstrates that in analogy to methods in human genetics, the SFS of viral sequences sampled well after latent infection can remain informative about the initial R0. Thus, it suggests the utility of analyzing the SFS of sequences derived from patient and animal trials of viral therapies, because in some cases, the initial R0 may be able to indicate subtle therapeutic progress, even in the absence of statistically significant differences in the infection of treatment and control groups.

Monotonous betting strategies in warped casinos

The principal result of this paper is the introduction of a readily verifiable sufficient condition for the existence of an invariant measure on transient Markov chains. Specifically, subject to mild additional regularity conditions, it is enough to check that for every compact set $A$ in the state space $X$, there exists a compact $C \supset A$ such that \begin{equation*}\tag{0.1} \lim_{y\in F, y\rightarrow\infty} \frac{P(y, A)}{P(y, C)} = 0\quad\text{if} \bar{F} \text{is not compact and} F = \{y: P(y, A) > 0\},\end{equation*} $P(y, A)$ denote the 1-step transition probabilities of the Markov chain. (0.1) is readily implied by simple conditions on the generating functional of the offspring distribution (Theorems 2.1 and 2.2) for discrete time multi-type Markov processes. These conditions together with some additional regularity-hypotheses are verified for (i) A 1-dimensional neutron branching model of Harris, (ii) Discrete-time age dependent branching processes, (iii) Galton-Watson processes with immigration. A generalization (to discrete-time temporally homogeneous Markov processes with $\sigma$-compact metric state space) of a condition of T. Harris (1957) for existence of invariant measures on transient Markov chains is also given. This condition is, unfortunately, difficult to check in specific examples. The sufficient condition (0.1) involves only 1-step transition probabilities, as opposed to the $n$-step transitions incorporated in the Harris condition, which is also necessary. Throughout this paper, all subsets of a topological space considered are assumed Borel measurable. The closure, interior, boundary and complement of a set $A$ are denoted by $\bar{A}, A^0, \partial A$ and $A^c$ respectively, and $I_A$ denotes the indicator function of $A$. The sets of positive integers, nonnegative integers, are denoted by $I$ and $I_0$ respectively. For any $\sigma$-compact metric space $X$, the following notations are used: $M(X) =$ set of regular measures (Borel measures finite on compact sets) on $X$. $B(X) =$ set of bounded measurable functions on $X$ with sup norm. $B'(X) = \{s \in B(X): \|s\| < 1\}$. $C(X) =$ set of bounded continuous functions on $X$. $C_{00}(X) = \{s\in C(X): s$ has compact support$\}$. $C_0(X) = \{s \in C(X):\lim_{x\rightarrow\infty}s(x) = 0\}.$ $C'(X) = C(X) \cap B'(X)$, similarly for $C_0'(X)$ and $C'_{00}(X)$. For any $\mathscr{L} \subset B(X), \mathscr{L}^+ = \{s \in \mathscr{L}: s \geqq 0\}$. If $\mu \in M(X)$ and $\in B(X)$, then $\mu f = \int_xf(x)\mu(dx)$ when defined.

We consider the supercritical bisexual Galton–Watson process\n(BGWP) with promiscuous mating, that is, a branching process which behaves like\nan ordinary supercritical Galton–Watson process (GWP) as long as at\nleast one male is born in each generation. For a certain example, it was\npointed out by Daley, Hull and Taylor [J. Appl. Probab. 23 (1986)\n585–600] that the extinction probability of such a BGWP apparently\nbehaves like a constant times the respective probability of its asexual\ncounterpart (where males do not matter) if the number of ancestors grows to\n$\\infty$. In an earlier paper, we provided general upper and lower bounds for\nthe ratio between both extinction probabilities and also numerical results that\nseemed to confirm the convergence of that ratio. However, theoretical\nconsiderations rather led us to the conjecture that this does not generally\nhold. The present article turns this conjecture into a rigorous result. The key\nstep in our analysis is to identify the extinction probability ratio as a\ncertain functional of a subcritical ordinary GWP and to prove its continuity as\na function of the number of ancestors in a suitable topology associated with\nthe entrance Martin boundary of that GWP.

Let {Zn} be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn/EZn}. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p &gt; 1, and which has an infinite number of rates of growth.

Invariance principle for random processes on Galton-Watson trees

Detection of cellular aging in a Galton–Watson process

Two closely related results are established. A given critical branching process in a random environment attains a high level and spends at that level a part of its life obeying the arcsine law. If a critical Galton–Watson process survives up to a distant moment, then the ratio of the total number of individuals of the future generations to the total number of individuals ever born in the process obeys the arcsine law.

Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different modelling assumptions, for instance different values of the number of types.

On sequential estimation for branching processes with immigration