Compound Poisson statistics for multiple returns in shrinking cylinders for mixing processes

2015 ◽  
Vol 36 (5) ◽  
pp. 1616-1643
Author(s):  
ARIEL RAPAPORT
Keyword(s):  
Periodic Point ◽  
Limit Distribution ◽  
Gibbs Measures ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Convergence In Distribution ◽  
Mixing Processes ◽  
Poisson Statistics ◽  
Global Condition ◽  
Necessary And Sufficient

Given a periodic point ${\it\omega}$ in a ${\it\psi}$-mixing shift with countable alphabet, the sequence $\{S_{n}\}$ of random variables counting the number of multiple returns in shrinking cylindrical neighborhoods of ${\it\omega}$ is considered. Necessary and sufficient conditions for the convergence in distribution of $\{S_{n}\}$ are obtained, and it is shown that the limit is a Pólya–Aeppli distribution. A global condition on the shift system which guarantees the convergence in distribution of $\{S_{n}\}$ for every periodic point is introduced. This condition is used to derive results for $f$-expansions and Gibbs measures. Results are also obtained concerning the possible limit distribution of sub-sequences $\{S_{n_{k}}\}$. A family of examples in which there is no convergence is presented. We also exhibit an example for which the limit distribution is pure Poissonian.

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

The probability that the largest observation is censored

1993 ◽  
Vol 30 (03) ◽  
pp. 602-615 ◽  
Author(s):  
R. A. Maller ◽  
S. Zhou
Keyword(s):  
Survival Time ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Consistent Estimator ◽  
Survival Times ◽  
Practical Difficulty ◽  
Independent Censoring ◽  
The Mean ◽  
Necessary And Sufficient

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.

Poisson theorem for non-homogeneous Markov chains

1989 ◽  
Vol 26 (03) ◽  
pp. 637-642 ◽  
Author(s):  
Janusz Pawłowski
Keyword(s):  
Markov Chains ◽  
Poisson Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compound Poisson Distribution ◽  
Convergence In Distribution ◽  
Compound Poisson ◽  
Necessary And Sufficient

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

The stochastic precedence ordering with applications in sampling and testing

2004 ◽  
Vol 41 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Philip J. Boland ◽  
Harshinder Singh ◽  
Bojan Cukic
Keyword(s):  
Random Sampling ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Simple Random Sampling ◽  
Order Type ◽  
Bernoulli Random Variables ◽  
Binomial Distributions ◽  
Necessary And Sufficient ◽  
Stochastic Precedence

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

scholarly journals The law of the iterated logarithm for exchangeable random variables

1995 ◽  
Vol 18 (2) ◽  
pp. 391-396
Author(s):  
Hu-Ming Zhang ◽  
Robert L. Taylor
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Exchangeable Random Variables ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient

In this note, necessary and sufficient conditions for laws of the iterated logarithm are developed for exchangeable random variables.

Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift

1986 ◽  
Vol 18 (04) ◽  
pp. 865-879 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Random Walks ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Partial Sums ◽  
Exit Times ◽  
First Passage ◽  
Partial Sum ◽  
Positive Expectation ◽  
Necessary And Sufficient

Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation. We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities.

The Hurst effect under trends

1983 ◽  
Vol 20 (3) ◽  
pp. 649-662 ◽  
Author(s):  
R. N. Bhattacharya ◽  
Vijay K. Gupta ◽  
Ed Waymire
Keyword(s):  
Hurst Exponent ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stationary Sequence ◽  
Monotonic Trend ◽  
Weakly Dependent Random Variables ◽  
Necessary And Sufficient ◽  
Hurst Effect ◽  
Weakly Dependent ◽  
Negative Parameter

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) = c(m + n)ß, where m is an arbitrary non-negative parameter and c is not 0. For – ½ < ß < 0 the Hurst exponent is shown to be precisely given by 1 + ß. For ß ≦ – ½ and for ß = 0 the Hurst exponent is 0.5, while for ß > 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.

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