# Showing papers in "Probability Theory and Related Fields in 1981"

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TL;DR: In this article, a probability density on an interval I, finite or infinite, including its finite endpoints, if any; and f vanishes outside of I. To define this object, choose a reference point xosI and a cell width h.

Abstract: Let f be a probability density on an interval I, finite or infinite: I includes its finite endpoints, if any; and f vanishes outside of I. Let X1, . . . ,X k be independent random variables, with common density f The empirical histogram for the X's is often used to estimate f To define this object, choose a reference point xosI and a cell width h. Let Nj be the number of X's falling in the j th class interval:

1,336 citations

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692 citations

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TL;DR: In this article, the strong law of large numbers for pairwise independent random variables has been shown to not use Kolmogorov's inequality, which is also the case for the weak law.

Abstract: In the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorov's inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].

301 citations

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TL;DR: In this paper, the authors consider a simple exclusion particle jump process on ℤ, where the underlying one particle motion is a degenerate random walk that moves only to the right, and prove that the distribution of the process looked at by an observer travelling at constant speed u, converges weakly to the Bernoulli measure with density f(u), as the time tends to infinity.

Abstract: One considers a simple exclusion particle jump process on ℤ, where the underlying one particle motion is a degenerate random walk that moves only to the right. One starts with the configuration in which the left halfline is completely occupied and the right one free. It is shown that the number of particles at time t between site [u t] and [v t], divided by t, converges a.s. to
$$\int\limits_u^
u {f(w)dw}$$
, where f might be called the density profile. It is explicitely determined and shown to be an affine function. Secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u, converges weakly to the Bernoulli measure with density f(u), as the time tends to infinity.

273 citations

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269 citations

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TL;DR: In this article, the convergence in probability of Malthus normed supercritical general branching processes (i.e., Crump-Mode-Jagers branching processes) with a general characteristic is established, provided the latter satisfies mild regularity conditions.

Abstract: Convergence in probability of Malthus normed supercritical general branching processes (i.e. Crump-Mode-Jagers branching processes) counted with a general characteristic are established, provided the latter satisfies mild regularity conditions. If the Laplace transform of the reproduction point process evaluated in the Malthusian parameter has a finite ‘x log x-moment’ convergence in probability of the empirical age distribution and more generally of the ratio of two differently counted versions of the process also follow.

269 citations

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TL;DR: In this article, the critical probabilities of site percolation on the square lattice satisfy the relation p c + p ✓ + p.............. +p.............. = 1 and the continuity of the function "percolation probability" is established.

Abstract: We prove that the critical probabilities of site percolation on the square lattice satisfy the relation p
c
+p
/*
=1. Furthermore we prove the continuity of the function “percolation probability”.

254 citations

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220 citations

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TL;DR: In this paper, the consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on Φ and χ and assuming the joint distribution of (x, y) and (y, y ) to fulfill the model (*) only approximately.

Abstract: Let (xini, y
i
be a sequence of independent identically distributed random variables, where x
i
∃R
p
and y
i
∃R, and let θ∃R
p
be an unknown vector such that y
i
=x′
i
θ+u
i
(*), where u
i
is independent of x
i
and has distribution function F(u/σ), where σ>0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, (θ
*,σ*), defined as solutions of the system:
$$\sum\limits_i \phi ({\text{x}}_i ,r_i )x_i = 0,\sum\limits_i \chi (|r_i |) = 0,$$
, where r= (y
i
−x
i
1θ*/σ)*, with Φ∶ R
p
×R→R and χ∶ R→R. This class contains estimators of (θ, σ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on Φ and χ and assuming the joint distribution of (x
i
, y
i
) to fulfill the model (*) only approximately.

125 citations

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TL;DR: In this article, the almost sure behaviour of the sample functions of the vector-valued N-parameter Wiener process and its stable analogues is investigated, especially their small fluctuations, and laws of the simple and of the iterated logarithm are established for the supremum of the local time increments or sojourn times.

Abstract: The almost sure behaviour of the sample functions of the vectorvalued N-parameter Wiener process and its stable analogues is investigated, especially their small fluctuations. In particular, laws of the simple and of the iterated logarithm are established for the supremum of the local time increments or sojourn times. These results give precise information about the minimum oscillation of the sample functions. In addition, the Hausdorff measure problem for the graph and the range of these processes is solved.

125 citations

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TL;DR: In this paper, it was shown that the product limit estimator of F is almost sure consistent with rate O(√log log n/n), that is P(sup ¦F====== *� �� ��(u)− F(u)-¦=0(1 −∞

__0, where T====== F>>\s �� =sup{x∶ F(x)>0}.__Abstract: Let X
1,X
2,...,X
n
be i.i.d. r.v.'-s with P(X>u)=F(u) and Y
1,Y
2,...,Y
n
be i.i.d. P(Y>u)=G(u) where both F and G are unknown continuous survival functions. For i=1,2,...,n set δ
i=1 if X
i
≦Y
i
and 0 if X
i
>y
i
, and Z
i
=min {itXi, Yi}. One way to estimate F from the observations (Z
i
,δ
i
) i=l,...,n is by means of the product limit (P.L.) estimator F
*
(Kaplan-Meier, 1958 [6]). In this paper it is shown that F
n
*
is uniformly almost sure consistent with rate O(√log logn/√n), that is P(sup ¦F
*
(u)− F(u)¦=0(√log log n/n)=1 −∞

__0, where T F =sup{x∶ F(x)>0}. A similar result is proved for the Bayesian estimator [9] of F. Moreover a sharpening of the exponential bound of [3] is given.__••

TL;DR: In this paper, the authors considered simple generalizations of the potlatch and smoothing processes which were introduced in [8] and studied in [5] and provided relatively simple examples of infinite interacting systems which exhibit phase transition.

Abstract: We consider simple generalizations of the potlatch and smoothing processes which were introduced in [8] and studied in [5]. These generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition. The original potlatch and smoothing processes do not exhibit phase transition. Our results show that for the generalized processes, phase transition does not usually occur in one or two dimensions, but usually does occur in higher dimensions. Upper and lower bounds for the relevant critical values are obtained. As one application of our results, we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d→∞, thus answering a question we raised in [3]. This is done by comparing the contact process with an appropriate generalized smoothing process.

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TL;DR: In this article, the authors consider the construction of infinite particle systems with locally interacting components and prove that the first and second moments of the number of particles per site in equilibrium can be computed explicitly.

Abstract: In [5] the second author introduced a variety of new infinite systems with locally interacting components. On the basis of computations for the finite analogues of these systems, he made conjectures ragarding their limiting behavior as t→∞. This paper is devoted to the construction of these processes and to the proofs of these conjectures. We restrict ourselves primarily to spatially homogeneous situations; interesting problems remain unsolved in inhomogeneous cases. Two features distinguish these processes from most other infinite particle systems which have been studied. One is that the state spaces of these systems are noncompact; the other that even though the invariant measures are not generally of product form, one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium. The second moment computations are of inherent interest of course, and they play an important role in the proofs of the ergodic theorems as well.

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TL;DR: In this article, a general scheme of random censorship can be defined in the following way: Let X be a real random variable with distribution function F(t) and X is a real variable whose observations have been randomly censored on the right.

Abstract: The empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample. It is not, however, appropriate when the observations are incomplete. This is the case, for example, when the observations have been randomly censored on the right. In the latter case the so-called product-limit estimate of Kaplan and Meier [16] has been generally accepted as a substitute for the empirical distribution function, and it possesses many of the same properties (cf. Meier [19]). Developing the corresponding statement of Efron [8], weak convergence, over a finite interval, of the Kaplan-Meier product-limit estimate to a Gaussian process when there is censorship on the right, has been obtained by Breslow and Crowley [3] (cf. also Meier [19]). Breslow and Crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship. A generalisation of the Efron-Breslow-Crowley theorem was formulated by Yang [24] for the case when censorship on the right is performed by more competing risks. These results have numerous statistical applications in areas such as medical follow-up studies, life testing, actuarial sciences and demography. Koziol and Green [18] have computed tables for the limiting distribution of the Cram6r-von Mises statistic based on the Kaplan-Meier product-limit estimate under a specific random censorship model. A point will be stressed in this paper, however, that by a certain transformation, (which seems to have escaped many researchers' attention and is due to Efron [8]), it is always possible to obtain the standard Wiener process in the limit, the distributions of many functionals of which are well known. This transformation was only recently used in a stimulating note by Gillespie and Fisher [14] to construct asymptotic confidence bands for the unknown distribution function in the Kaplan-Meier model. A general scheme of random censorship can be defined in the following way: Let X be a real random variable with distribution function F(t)

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TL;DR: In this paper, a flexible approach to proving the Central Limit Theorem (C.L.T) for triangular arrays of dependent random variables (r.v.s) was given.

Abstract: This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak ‘mixing’ condition called l-mixing. Roughly speaking, an array of real r.v.s is said to be l-mixing if linear combinations of its ‘past’ and ‘future’ are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, ϱ-mixing and ψ-mixing) are special cases of l-mixing. Linear processes are shown to be l-mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are ‘nearly second order stationary’.

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TL;DR: In this paper, the asymptotic behavior of g2n and g3nis was studied utilizing the properties of the new estimator g1n, and consistency results were established and rates of uniform convergence obtained.

Abstract: Consider the regression model Yi*=g(xi*)+ei*, i=1,2,...,n, where xi*'s denote unordered design variables, and g is an unknown function defined on the interval [0,1]. Assume {ei*} are iid random variables with zero mean and finite variance. Priestley and Chao (1972) and Clark (1977) proposed estimators g2nand g3n, respectively for g. In this paper, the asymptotic behavior of g2nand g3nis studied utilizing the properties of the new estimator g1n. It is shown that g1n, g2n, g3nare asymptotically equivalent in various senses. Moreover, consistency results are established and rates of uniform convergence obtained. For example, if E¦e*¦3<∞, if g is Lipschitz of order 1, and if {Βn} is any sequence of constants tending to ∞ as n→∞, then for all \(0 < a \leqq b < 1,({{n^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-
ulldelimiterspace} 3}} } \mathord{\left/ {\vphantom {{n^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-
ulldelimiterspace} 3}} } {\beta _n \log n)\mathop {\sup }\limits_{a \leqq x \leqq b} |g_{1n} (x) - g(x)|\xrightarrow{{w.p.1}}0,}}} \right. \kern-
ulldelimiterspace} {\beta _n \log n)\mathop {\sup }\limits_{a \leqq x \leqq b} |g_{1n} (x) - g(x)|\xrightarrow{{w.p.1}}0,}}\), as n→∞. Finally, when g is monotone, a strong consistent isotonic estimator gn*is considered.

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TL;DR: In this article, a stochastic differential equation with smooth coefficients is considered, which defines a continuous flow φt, (ω,.) of C+8 mappings of Rd in Rd.

Abstract: A stochastic differential equation with smooth coefficients is considered, which defines a continuous flow φt, (ω, .) of C+8 mappings of Rd in Rd. If zt is a continuous semi-martingale, φt,(ω,zt)s> is proved to be a semi-martingale, for which an Ito type formula is established. It is shown that a.s., for any t,φt(ω, .) is an onto diffeomorphism. If zt is a continuous semi-martingale, φt−1,(ω,zt) is proved to be a semi-martingale, whose Ito decomposition is explicitly found.

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TL;DR: In this paper, it was shown that the stationary distribution of the queue length at arrivals has an exact geometric tail of rate η, 0 0.1, ξ, η and ξ may be evaluated together by an elementary algorithm.

Abstract: This paper deals with the stablec-server queue with renewal input. The service time distributions may be different for the various servers. They are however all probability distributions of phase type. It is shown that the stationary distribution of the queue length at arrivals has an exact geometric tail of rate η, 0 0. The quantities η and ξ may be evaluated together by an elementary algorithm. For both distributions, the multiplicative constants which arise in the asymptotic forms may be fully characterized. These constants are however difficult to compute in general.

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TL;DR: It is proved that for fixed A~, . . . ,A, the bounds in (1.1) are attained and sharp upper and lower bounds for {~p d P; PeJ ( (P 1 .. . . .) P,)} for more general functions ~o on W are derived.

Abstract: where for a~R 1, a+ =max{a,0}. Though very simple the bounds in (1.1) are useful in many applications (cf. [5, 11, 16, 14]). In the present paper we prove that for fixed A~, . . . ,A, the bounds in (1.1) are attained. Furthermore, we shall derive sharp upper and lower bounds for {~p d P; PeJ ( (P 1 .. . . . P,)} for more general functions ~o on W. In the special case that ~ = { 0 , 1 } , l< i

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TL;DR: In this article, it was shown that the Cramer-von-mises estimator is robust to contaminations of various sorts, and that the data actually collected by the statistician follow a distribution that is close to, but possibly distinct from, the P θ..............'s. And it was proved that, under these conditions and in an appropriate asymptotic framework, the minimum distance estimators of the Mises type are robust.

Abstract: Given a fixed parametric family {P
θ
} it is desired to estimateθ. However, due to contaminations of various sorts, the data actually collected by the statistician follow a distribution that is close to, but possibly distinct from, theP
θ
's. It is proved that, under these conditions and in an appropriate asymptotic framework, the minimum distance estimators of the Cramer-von Mises type are robust. Specification of a Cramer-von Mises weight functionH defines a notion of distance; each such choice ofH then delineates the kind of contamination possible, and leads to an estimator which defends optimally against it. When the theory is specialized to location models, various choices ofH lead to estimators asymptotically equivalent to such familiar ones as trimmed mean, median, Hodges-Lehmann estimator, and so forth. The framework developed herein provides some guidelines for choosing among the possible estimators, and suggests that the standard Cramer-von Mises estimator of location is probably as good a robust estimator as any.

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TL;DR: Adaptive stochastic approximation schemes for choosing the levels of x at which y is to be observed are useful in applications of the following nature: as discussed by the authors assume that in (1.1) x is the dosage level of a drug given to the i-th patient who turns up for treatment and that Yi is the response of the patient.

Abstract: Adaptive stochastic approximation schemes for choosing the levels of x at which y is to be observed are useful in applications of the following nature. Suppose that in (1.1) x~ is the dosage level of a drug given to the i-th patient who turns up for treatment and that Yi is the response of the patient. Suppose also that an optimal response value y* is desired. Without loss of generality, we shall (replacing y~ by y~-y* if necessary) assume that y*=0. If 0 were known, then the dosage levels should all be set at 0. Since 0 is usually unknown, how can the dosage levels xi be chosen so as to approach the

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TL;DR: In this article, it was shown that the existence of the asymptotically invariant sequence of probabilities in the hypothesis of the Hunt-Stein theorem is equivalent to amenability, a condition that has been much studied by functional analysts.

Abstract: A number of conditions on groups have appeared in the literature of invariant statistical models in connection with minimaxity, approximation of invariant Bayes priors by proper priors, the relationship between Bayesian and classical inference, ergodic theorems, and other matters. In the last decade, rapid development has occurred in the field and many of these conditions are now known to be equivalent. We survey the subject, make the equivalences explicit, and list some groups of statistical interest which do, and also some which do not, have these properties. In particular, it is shown that the existence of the asymptotically invariant sequence of probabilities in the hypothesis of the Hunt-Stein theorem is equivalent to amenability, a condition that has been much studied by functional analysts.

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TL;DR: In this article, the first exit densities of Brownian motion with drift are derived for upper and lower class functions, leading to uniform approximations for the power functions of sequential tests.

Abstract: Let {ψa; a e ℝ} be a sequence of curved boundaries which tend to infinity as a increases. Let
$$T_a = \inf \{ t > 0|W(t) \geqq \psi _a (t)\} $$
where W(t) denotes the standard Brownian motion. Under regularity conditions on the boundaries uniform approximations for the first exit densities of Ta are derived. The consequences for upper and lower class functions are discussed. The approximations for the first exit densities of Brownian motion with drift, which are also derived, lead to uniform approximations for the power functions of sequential tests. The quality of the approximations is demonstrated by some figures.

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TL;DR: In this article, the authors established a law of the iterated logarithm for a triangular array of independent random variables, and applied it to obtain laws for a large class of nonparametric density estimators.

Abstract: We establish a law of the iterated logarithm for a triangular array of independent random variables, and apply it to obtain laws for a large class of nonparametric density estimators. We consider the case of Rosenblatt-Parzen kernel estimators, trigonometric series estimators and orthogonal polynomial estimators in detail, and point out that our technique has wider application.

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TL;DR: In this article, limit theorems with a non-Gaussian (in fact nonstable) limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence.

Abstract: Limit theorems with a non-Gaussian (in fact nonstable) limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. Analogous limit theorems are here obtained for finite Fourier transforms of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence.