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The paper begins with an iterated logarithm law of classical Hartman-Wintner form for stationary martingales. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters in a stationary autoregressive process. In the case of an autore-gression of small order, detailed rate results for each autocorrelation and for the estimators...
This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem, Mertens’ theorem and the Riemann hypothesis have previously been established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results for both martingales and tail sums of martingale differences...
This paper is concerned with the estimation of a parameter of a stochastic process on the basis of a single realization. It is shown, under suitable regularity conditions, that the maximum likelihood estimator is the best consistent asymptotically normal estimator in the sense of having minimum asymptotic variance. It also produces the best limiting probability of concentration in symmetric intervals...
Summary In this paper, the precise asymptotic behaviour of the large deviation probability is found in the case where the random variables are attracted to a non-normal stable law. This extends previous work of the same author in which only the order of magnitude of the large deviation probability was found.
It has recently emerged that the central limit theorem and iterated logarithm law for random walk processes have natural counterparts for Galton-Watson processes with or without immigration. Much of the work on these counterparts has previously involved the imposition of supplementary moment conditions. In this paper we show how to dispense with these supplementary conditions and in so doing make...
Summary Asymptotic normality of the posterior distribution of a parameter in a stochastic process is shown to hold under conditions which do little more than ensure consistency of a maximum likelihood estimator. Much more stringent conditions are required to ensure asymptotic normality of the MLE. This contrast, which has implications of considerable significance, does not emerge in the classical...
Let Xii = 1, 2, 3,… be a sequence of independent and identically distributed random variables with EXi = o and varXi = l. Write F(x) for the distribution function and f(t) for the characteristic function of Xi and put $$S_n = \sum\limits_{i = 1}^n {X_i } $$ . Then, $$F_n(x) = P(S_n \leqq x\sqrt{n})\rightarrow \Phi(x) = \frac{1}{\sqrt{2\pi}}...
Estimation of parameters in diffusion models is usually handled by maximum likelihood and involves the calculation of a Radon-Nikodym derivative. This methodology is often not available when minor changes are made to the model. However, these complications can usually be avoided and results obtained under more general conditions using quasi-likelihood methods. The basic ideas are explained in this...
Let {Xii = 1, 2, 3, …} be a sequence of independent and identically distributed random variables for which EXi = 0, var Xi = σ2, E|Xi|2 +a = b < ∞, a > 0. C[0, 1] is the space of continuous real-valued functions on the interval...
Let Z0 = 1, Z1, Z2... denote a super-critical Galton–Watson process whose non-degenerate offspring distribution has probability generating function $$F\left(s\right)=\sum\nolimits^{\infty}_{{j}=0}s^j \mathbf{P}{\rm r}\left(\mathbf{Z}_1=j\right), 0\leqq s \leqq 1,$$ where 1 <m =EZ1 >∞. The Galton–Watson process evolves in such that the generating...
Summary This note pertains to a generalized model for random fluctuation of allele frequency, where the population size is permitted to fluctuate randomly from generation to generation. Martingale methods are applied to discuss in two propositions, respectively, necessary and sufficient conditions for P (Y(11−Y)>0)>0, where Y is the (almost sure) limiting frequency of one allele. Under an additional...
This paper is concerned with methodology for studying the long-term genetic composition of a population of haploid individuals in the case where the population size is varying. A general approach requiring a minimum of assumptions is described based on constructing martingales out of expressions for the means of the numbers of allelic types, conditional on the past. Earlier investigations were based...
Let $$X_i, i = 1, 2, 3, \ldots,$$ be a sequence of independent and identically distributed random variables with law $$\mathcal{L}(X)$$ and write $$S_n = \sum\nolimits^{n}_{i=1} X_i$$ . Let $$x_n, n = 1, 2, 3, \ldots,$$ be a monotone sequence of positive numbers with $$x_n \rightarrow \infty \,{\rm as}\, n \rightarrow \infty$$ such that $$x_n^{-1} S_n \rightarrow_P 0$$...
Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing...
There are two basic parts to this paper. In the first part we suppose that Xi, i=1,2,3,… is a sequence of independent and identically distributed random variables. We write $$S_n = \sum\limits_{i = 1}^n {X_i,n \geqq 1} $$ , and suppose that the variables are centered so that EXi = 0 if E ǀXi...
In this paper it is shown how martingale theorems can be used to appreciably widen the scope of classical inferential results concerning autocorrelations in time series analysis. The object of study is a process which is basically the second-order stationary purely non-deterministic process and contains, in particular, the mixed autoregressive and moving average process. We obtain a strong law and...
Gene survival in a population which increases without density dependence is considered using a generalization of the Moran model for haploid individuals. It is shown that situations where ultimate homozygosity is certain and where there is a non-zero probability of balanced polymophism are both possible. Necessary and sufficient conditions in terms of the mean of the population growth distribution...
Let Z0 = l, Z1,Z2, … denote a super-critical Galton-Watson branching process with 1 <EZ1 =m <∞ and 0 <varZ1 = σ2 < ∞It is well-known that there exists a non-degenerate random variable W such that $$\mathop {\lim }\limits_{n \to \infty } W_n = W$$ almost surely, where Wn = m−n...
Let $$\{X_i, i=1, 2, 3, \ldots\}$$ be a sequence of independent and identically distributed random variables and write $$S_n=\sum\nolimits^n_{i=1} X_i,\: n\geqq1.$$ It is well known that $$\Pr ({\mathop {\lim \sup }\limits_{n \to \infty } \!({2 n\:\log \:\log \:n})^{ - {1}/ {2}} \!{S_n } = 1}) = 1$$ If and only if $$EX_i=0,\: EX^2_i=1$$ (Hartman and Wintner [6] obtained...
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