Showing papers in "Probability Theory and Related Fields in 1983"
TL;DR: In this paper, an extension of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction: a weak mixing condition restricting long range dependence and a local condition restricting the clustering of high level exceedances.
Abstract: Extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction:
(a)
a weak “mixing condition” restricting long range dependence
(b)
a local condition restricting the “clustering” of high level exceedances.
436 citations
TL;DR: In this article, it was shown that the generalized potlatch and smoothing process has a nontrivial fixed point in the special case that the variables are independent and identically distributed, or are fixed multiples of one random variable.
Abstract: Let W
1,..., W
N
be N nonnegative random variables and let
$$\mathfrak{M}$$
be the class of all probability measures on [0, ∞). Define a transformation T on
$$\mathfrak{M}$$
by letting Tμ be the distribution of W
1X1+ ... + W
N
X
N
, where the X
i
are independent random variables with distribution μ, which are independent of W
1,..., W
N
as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the W
i
are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some γ>1, EW
<∞ for all i, we determine exactly when T has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, T always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett.
298 citations
TL;DR: In this paper, the authors investigated the relation between the speed of estimation and the metric structure of the parameter space Θ, especially in the case when its metric dimension is infinite and gave a construction for some sort of universal estimates the risk of which is bounded by C 2 r q(n) in all cases where the preceding theory applies.
Abstract: We investigate the relations between the speed of estimation and the metric structure of the parameter space Θ, especially in the case when its metric dimension is infinite. Given some distance d on Θ (generally Hellinger distance in the case of n i.i.d. variables), we consider the minimax risk for n observations: \(R_n (q) = \mathop {\inf \sup }\limits_{T_{n } \theta \in \Theta } {\text{IE}}_\theta [d^q (\theta , T_n )],\), T n being any estimate of θ. We shall look for functions r such that for positive constants C 1(q) and C 2(q) C 1 r q(n)≦R n(q)≦C2 r q(n). r(n) is the speed of estimation and we shall show under fairly general conditions (including i.i.d. variables and regular cases of Markov chains and stationnary gaussian processes) that r(n) is determined, up to multiplicative constants, by the metric structure of Θ. We shall also give a construction for some sort of “universal” estimates the risk of which is bounded by C 2 r q(n) in all cases where the preceding theory applies.
283 citations
TL;DR: In this article, the authors extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorem and as approximation by a Brownian motion.
Abstract: Some probabilistic limit theorems for Hoeffding's U-statistic [13] and v Mises' functional are established when the underlying processes are not necessarily independent We consider absolutely regular processes [24] and processes (X
n)n≧1 which are uniformly mixing [14] as well as their time reversal (X
−n
)n≦−1, called uniformly mixing in both directions of time Many authors have weakened the hypothesis of independence in statistical limit theorems and considered m-dependent, Markov or weakly dependent processes; in particular for U statistics under weak dependence Sen [22] has considered *-mixing processes and derived a central limit theorem and a law of the iterated logarithm, while Yoshihara [26] proved central limit theorems and as results in the absolutely regular and uniformly mixing case Here we extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseen type theorem), functional central limit theorems and as approximation by a Brownian motion Extensions to multisample versions and other extensions are briefly discussed
231 citations
TL;DR: In this article, it was shown that formal Edgeworth expansions are valid for sums of weakly dependent random vectors and the error of approximation has ordero(n−(s−2)/2) if the strong mixing coefficients in (iii) are decreasing at an exponential rate.
Abstract: It is shown that formal Edgeworth expansions are valid for sums of weakly dependent random vectors. The error of approximation has ordero(n
−(s−2)/2) if
The strong mixing coefficients in (iii) are decreasing at an exponential rate. The above conditions can easily be checked and are often satisfied when the sequence of random vectors is a Gaussian, or a Markov, or an autoregressive process. Explicit formulas are given for the distribution of finite Fourier transforms of a strictly stationary time series.
208 citations
166 citations
TL;DR: In this paper, almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space, and it is then shown that empirical processes readily fit into this general framework.
Abstract: Almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space. It is then shown that empirical processes readily fit into this general framework. Thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes.
145 citations
TL;DR: In this article, the authors developed a stochastic calculus of variations for strong Markov processes with jumps, which is the analogous of the Malliavin calculus for variations on diffusions.
Abstract: The purpose of this paper is to develop a stochastic calculus of variations for R
n
-valuedstrong Markov processes with jumps x
t
,which is the analogous of the Malliavin calculus of variations on diffusions An integration by parts formula is established on a non Gaussian infinite dimensional probability space, in order to prove regularity of the probability law on R
n
of x
t
,for fixed time t Diffusions with jumps are also considered The connection between the calculus of variations and the representations of martingales for jump process is exhibited
131 citations
TL;DR: In this paper, the maximal deviation of the product-limit estimate from the estimated distribution function is investigated, and the log log law is proved for this deviation on appropriate half lines, with the precise constants.
Abstract: The maximal deviation of the product-limit estimate from the estimated distribution function is investigated. As a consequence of a functional law of the iterated logarithm, the log log law is proved for this deviation on appropriate half lines, with the precise constants. This result implies that the log log law need not hold in general for the maximal deviation on the whole line. Then a general asymptotic order of magnitude is determined for the latter deviation. This order depends on the joint behaviour of the censoring and censored distributions in a well-defined way. Corresponding to specific joint behaviours, several limsup results are deduced as consequence including all the previously known log log-type laws in improved form. The results are also extended to the variable censoring model.
114 citations
TL;DR: In this article, the invariance principle for ϕ-mixing sequences is investigated, satisfying restrictions on the variances which are a weak form of stationarity, where no mixing rate is assumed.
Abstract: In this paper we investigate the invariance principle for ϕ-mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For ϕ-mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.
93 citations
TL;DR: In this article, the authors consider a random walk on a finite group G and show that the time taken for the transition probabilities to approach the uniform distribution is small compared with #G, where R is the mean number of returns to the initial state in the short term.
Abstract: Consider a random walk on a finite group G. Suppose that the time taken for the transition probabilities to approach the uniform distribution is small compared with #G. Then the time taken for the random walk to visit every state is approximately R#G·log(#G), where R is the mean number of returns to the initial state in the short term.
TL;DR: For strong mixing sequences with error term o((t log log logt)1/2) as mentioned in this paper, a strong invariance principle was obtained by Berkes and Philipp (1979).
Abstract: We prove an estimate for the Prohorov-distance in the central limit theorem for strong mixing Banach space valued random variables. Using a recent variant of an approximation theorem of Berkes and Philipp (1979) we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term
$$t^{\tfrac{1}{2} - \gamma }$$
. For strong mixing sequences we prove a strong invariance principle with error term o((t log logt)1/2).
TL;DR: In this paper, the Lipschitz-1/2 modulus of U n is defined by the first n of a sequence ξ 1, ξ 2 of uniform (0, 1) random variables.
Abstract: Let \(U_n (t) = n^{\tfrac{1}{2}} (\Gamma _n (t) - t), 0 \leqq t \leqq 1\), denote the uniform empirical process based on the first n of a sequence ξ 1, ξ 2, ... of iid uniform (0,1) random variables where \(\Gamma _n (t) = n^{ - 1} \sum\limits_{i = 1}^n {1_{[0,t]} } (\xi _i )\) is the empirical distribution function. The oscillation modulus of U n is defined by
$$\omega _n (a) = sup \{ |U_n (t + h) - U_n (t)|: 0 \leqq t \leqq 1 - h, h \leqq a\}$$
, and the Lipschitz-1/2 modulus of U n is defined by
$$\tilde \omega _n (a) = sup \{ |U_n (t + h) - U_n (t)|/h^{\tfrac{1}{2}} : 0 \leqq t \leqq 1 - h, a \leqq h \leqq 1\}$$
TL;DR: In this article, it was shown that for appropriate Skorohod construction, the convergence rate of the Wiener process is O(n^{1 \kern-
ulldelimiterspace} {4 + \varepsilon }}} ){\text{a.i.d.s.
Abstract: Let Y
1
, Y
2
, ... be a sequence of i.i.d. random variables with distribution P(Y
1 = k) = p
k
(k = ±1, ±2,...), E(Y
1) = 0, E(Y
1
2
) = σ2<∞. Put T
n
= Y
1+...+Y
n
and N(x,n) = # {k:0
TL;DR: In this article, the authors investigated the limit theorem for L n, the location of the rightmost particle in a supercritical branching random walk, under the assumption that the offspring distribution has ϕ(θ) = ∝ exp( √ n/a n)dF(x) 0 and showed that L n /a n converges weakly to a nondegenerate distribution.
Abstract: In recent years several authors have obtained limit theorems for L n , the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution has ϕ(θ) = ∝ exp(θx)dF(x) 0 In this paper we investigate what happens when there is a slowly varying function K so that 1−F(x)∼x }-q K(x) as x → ∞ and log(−x)F(x)→0 as x→−∞ In this case we find that there is a sequence of constants a n , which grow exponentially, so that L n /a n converges weakly to a nondegenerate distribution This result is in sharp contrast to the linear growth of L n observed in the case ϕ(θ)<∞
TL;DR: In this paper, the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2) were shown to be equivalent.
Abstract: For B a type 2 Banach lattice, we obtain a relationship between the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2). We then obtain some two-sided estimates for E∥Sn∥pwhich in lpspaces, 1≦p<∞, give n.a.s.c. for the weak law of large numbers. As a consequence of these estimates we also solve the domain of attraction problem in lp, p<2. Several examples and counterexamples are provided.
TL;DR: In this article, the Hausdorff dimension of the two-sided slow points of the Brownian path has been shown to be O(n −1/2, 1/2, x ≥ 1.3069, where M is the confluent hypergeometric function.
Abstract: For each −∞≦c
1 0,{\text{ }}B(t + h) - B(t) \in [c_1 h^{1/2} ,c_2 h^{1/2} ]\forall 0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } h\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \Delta \} .$$
. We find the Hausdorff dimension of S
+(c
1, c
2)(ω) as well as S
+(c
1, c
2)∩A(ω), where A a progressively measurable set satisfying certain hypotheses. In particular, the Hausdorff dimension of the two-sided slow points, S(c
1, c
2), and the (one-sided and two-sided) slow points in the zero set of B are obtained by making an appropriate choice of A. These results show, for example, that S(−c,c)≠O or = O, according as c is greater than or less than the smallest positive zero of M(−1/2, 1/2, x
2/2) (≅ 1.3069), where M is the confluent hypergeometric function. This confirms a conjecture of Burgess Davis, whose methods play a major role in the proofs. The results are also extended to higher dimensions.
TL;DR: In this paper, under the conditions related to moment and dependence coefficients, it was shown that the central limit theorem for random fields satisfying some weak dependence condition has a rate of O(¦V¦−1/2(log σ)d) (m-dependent case).
Abstract: Let {X
a
;a∈Z
d} (d≧2) be a random field satisfying some weak dependence condition. For a finite subset V of Z
d
, set
$$S(V) = \sum\limits_{a \in V} {X_a } $$
. In this paper, under the conditions related to moment and dependence coefficients, we show that L
∞- and L
1-rates in the central limit theorem for S(V) are of order O(¦V¦−1/2(log¦V¦)d) (strong mixing case): O(¦V¦−1/2) (m-dependent case). Here ¦V¦ denotes the number of elements in V. The content of this paper is a negative answer to the conjecture of Prakasa Rao (Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 247–256 (1981)).
TL;DR: In this article, the authors propose a method to solve the problem of unstructured data in order to improve the quality of the data collected by the data collection system, which can be found in Table 1.
Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
TL;DR: In this paper, a necessary condition is given for a finitary isomorphism between mixing Markov shifts of equal entropy to have finite expected coding time, and a necessary and sufficient condition is also given for such isomorphisms to be finitary.
Abstract: A necessary condition is given for a finitary isomorphism between mixing Markov shifts of equal entropy to have finite expected coding time.
TL;DR: In this article, the authors prove that if a ball in R is a Borel function on it and the Borel functions on it are Borels, then MathType!MTEF!2!1!+
Abstract: Let B be a ball in R
d and q be a Borel function on it. We prove that if
% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpe0-% rq0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaGaci% 4CaiaacwhacaGGWbaaleaacaWG4bGaeyicI4SabmOqayaaraaabeaa% ieaakiaa-bcadaWdrbqaamaalaaabaGaaiiFaiaadghacaGGOaGaam% yEaiaacMcacaGG8baabaGaaiiFaiaadIhacqGHsislcaWG5bGaaiiF% amaaCaaaleqabaGaamizaiabgkHiTiaaikdaaaaaaOGaa8hiaGqaci% aa+rgacaGF5baaleaacaWGcbaabeqdcqGHRiI8aaaa!4FB6!
$$\mathop {\sup }\limits_{x \in \bar B} \int\limits_B {\frac{{|q(y)|}}{{|x - y|^{d - 2} }} dy}$$
is sma11 enough, then
% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpe0-% rq0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxababaGaci% 4CaiaacwhacaGGWbaaleaadaWfqaqaaiaadIhacqGHiiIZcaWGcbaa% meaacaWG6bGaeyicI4SaeyOaIyRaamOqaaqabaaaleqaaGqaaOGaa8% hiaGqaciaa+veadaqhaaWcbaGaa4NEaaqaaiaa+HhaaaGcdaWadaqa% aiGacwgacaGG4bGaaiiCaiaa-bcadaWdXbqaaiaa+fhacaWFOaGaa4% hEamaaBaaaleaacaGF0baabeaakiaacMcacaWGKbGaamiDaaWcbaGa% a8hmaaqaaiaadshadaWgaaadbaGaamOqaaqabaaaniabgUIiYdaaki% aawUfacaGLDbaacaWFGaGaa8hpaiaa-bcacaWFRaGaa8hiaiabg6Hi% Lcaa!5A6B!
$$\mathop {\sup }\limits_{\mathop {x \in B}\limits_{z \in \partial B} } E_z^x \left[ {\exp \int\limits_0^{t_B } {q(x_t )dt} } \right] < + \infty$$
This paper gives a new proof of one of the two main results by Aizenman and Simon in [1] by a simple and elementary method. A basic theorem in Chung and Rao [2] is extended to the class of q treated in [1].
TL;DR: The first-exit time of a Brownian motion over the one-sided moving boundary given by x = f(t) is the density of the first exit time as discussed by the authors.
Abstract: Let p(t) be the density of the first-exit time of a Brownian motion over the one-sided moving boundary given by x=f(t). We derive the following formal expansion for p:
$$p(t) \sim \varphi (t^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} f(t))\left[ {t^{ - {\raise0.5ex\hbox{$\scriptstyle 3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \lambda (t) - \sum\limits_{n = 1}^\infty {c_n } t^{{\raise0.5ex\hbox{$\scriptstyle n$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} m_n (t^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \lambda (t))} \right]$$
Here λ(t)=f(t)−f′(t), ϕ is the standard normal density, m
n is the Hermite function of order (−n), and the coefficients c
n are functions of the derivatives of f at t. We give bounds for the error incurred by approximating p by the first n terms of the series, and examples in which the series provides an asymptotic expansion for p.