Random Variables Xn Research Articles

Central limit theorem for the number of records in Fα-scheme

Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.

Near-Record Values in Discrete Random Sequences

Given a sequence (Xn) of random variables, Xn is said to be a near-record if Xn∈(Mn−1−a,Mn−1], where Mn=max{X1,…,Xn} and a>0 is a parameter. We investigate the point process η on [0,∞) of near-record values from an integer-valued, independent and identically distributed sequence, showing that it is a Bernoulli cluster process. We derive the probability generating functional of η and formulas for the expectation, variance and covariance of the counting variables η(A),A⊂[0,∞). We also derive the strong convergence and asymptotic normality of η([0,n]), as n→∞, under mild regularity conditions on the distribution of the observations. For heavy-tailed distributions, with square-summable hazard rates, we prove that η([0,n]) grows to a finite random limit and compute its probability generating function. We present examples of the application of our results to particular distributions, covering a wide range of behaviours in terms of their right tails.

Compound Poisson approximation

We overview the results on the topic of compound Poisson approximation to the distribution of a sum Sn=X1+⋯+Xn of (possibly dependent) random variables. We indicate a number of open problems and discuss directions of further research.

A Central Limit Theorem for Predictive Distributions

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(·)=P(Xn+1∈·∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn=Dn(f):f∈F stably converge to a Gaussian kernel with a known covariance structure. In addition, Eφ(Dn(f))∣X1,…,Xn converges in probability for all f∈F and φ∈Cb(R).

Finite automata, probabilistic method, and occurrence enumeration of a pattern in words and permutations.

The main theme of this paper is the enumeration of the order-isomorphic occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence , the number of n-array k-ary words that contain a given pattern v exactly r times. In addition, we study the asymptotic behavior of the random variable Xn , the number of pattern occurrences in a random n-array word. The two topics are closely related through the identity . In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence converges to a limit independent of r, and determine the value of the limit. We then obtain several limit theorems for the distribution of Xn , including a central limit theorem, large deviation estimates, and the exact growth rate of the entropy of Xn . Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations.

Central limit theorems from the roots of probability generating functions

For each n, let Xn∈{0,…,n} be a random variable with mean μn, standard deviation σn, and letPn(z)=∑k=0nP(Xn=k)zk, be its probability generating function. We show that if none of the complex zeros of the polynomials {Pn(z)} is contained in a neighborhood of 1∈C and σn>nε for some ε>0, then Xn⁎=(Xn−μn)σn−1 is asymptotically normal as n→∞: that is, it tends in distribution to a random variable Z∼N(0,1). On the other hand, we show that there exist sequences of random variables {Xn} with σn>Clog⁡n for which Pn(z) has no roots near 1 and Xn⁎ is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of Pn(z) and the distribution of the random variable Xn.

Asymptotic normality in the problem of selfish parking

We continue in this paper to study one of the models of a discrete analog of the Renyi problem, also known as the parking problem. Suppose that n and i are integers satisfying n ≥ 0 and 0 ≤ i ≤ n – 1. We place an open interval (i, i + 1) in the segment [0, n] with i being a random variable taking values 0, 1, 2, …, n – 1 with equal probability for all n ≥ 2. If n < 2, then we say that the interval does not fit. After placing the first interval, two free segments [0, i] and [i + 1, n] are formed and independently filled with intervals of unit length according to the same rule, and so on. At the end of the process of filling the segment [0, n] with intervals of unit length, the distance between any two adjacent unit intervals does not exceed one. Suppose now that Xn is the number of unit intervals placed. In our earlier work published in 2018, we studied the asymptotic behavior of the first moments of random variable Xn. In contrast to the classical case, the exact expressions for the expectation, variance, and third central moment were obtained. The asymptotic behavior of all central moments of random variable Xn is investigated in this paper and the asymptotic normality for Xn is proved.

Partial Autoinformation to Characterize Symbolic Sequences.

An information-theoretic approach to numerically determine the Markov order of discrete stochastic processes defined over a finite state space is introduced. To measure statistical dependencies between different time points of symbolic time series, two information-theoretic measures are proposed. The first measure is time-lagged mutual information between the random variables Xn and Xn+k, representing the values of the process at time points n and n + k, respectively. The measure will be termed autoinformation, in analogy to the autocorrelation function for metric time series, but using Shannon entropy rather than linear correlation. This measure is complemented by the conditional mutual information between Xn and Xn+k, removing the influence of the intermediate values Xn+k−1, …, Xn+1. The second measure is termed partial autoinformation, in analogy to the partial autocorrelation function (PACF) in metric time series analysis. Mathematical relations with known quantities such as the entropy rate and active information storage are established. Both measures are applied to a number of examples, ranging from theoretical Markov and non-Markov processes with known stochastic properties, to models from statistical physics, and finally, to a discrete transform of an EEG data set. The combination of autoinformation and partial autoinformation yields important insights into the temporal structure of the data in all test cases. For first- and higher-order Markov processes, partial autoinformation correctly identifies the order parameter, but also suggests extended, non-Markovian effects in the examples that lack the Markov property. For three hidden Markov models (HMMs), the underlying Markov order is found. The combination of both quantities may be used as an early step in the analysis of experimental, non-metric time series and can be employed to discover higher-order Markov dependencies, non-Markovianity and periodicities in symbolic time series.

The Problem of Selfish Parking

One of the models of discrete analog of the Renyi problem known as the “parking problem” has been considered. Let n and i be integers, n ≥ 0, and 0 ≤ i ≤ n–1. Open interval (i, i + 1), where i is a random variable taking values 0, 1, 2, …, and n–1 for all n ≥ 2 with equal probability, is placed on interval [0, n]. If n < 2, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + 1, n] are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. When the filling of [0, n] with unit intervals is completed, the distance between any two neighboring intervals does not exceed 1. Let Xn be the number of placed intervals. This paper analyzes the asymptotic behavior of moments of random variable Xn. Unlike the classical case, exact expressions for the first moments can be found.

Discretizing Malliavin calculus

Suppose B is a Brownian motion and Bn is an approximating sequence of rescaled random walks on the same probability space converging to B pointwise in probability. We provide necessary and sufficient conditions for weak and strong L2-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence (Xn) of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to Bn, we derive necessary and sufficient conditions for strong L2-convergence to a σ(B)-measurable random variable X via convergence of the discrete chaos coefficients of Xn to the continuous chaos coefficients.

Products of random variables and the first digit phenomenon

We provide conditions on dependent and on non-stationary random variables Xn ensuring that the mantissa of the sequence of products ∏1nXk is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables.

Functional weak convergence of partial maxima processes

For a strictly stationary sequence of nonnegative regularly varying random variables (X n ) we study functional weak convergence of partial maxima processes $M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]$ in the space D[0, 1] with the Skorohod J 1 topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J 1 and M 1 functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition $\mathcal {A}(a_{n})$ with the time component.

On strong large deviation results for lightly trimmed sums and some applications

We obtain here the large deviation results for trimmed sums ((r)Sn) of i.i.d. random variables (Xn), with the distribution function belonging to the domain of attraction of a positive stable law. As an application, we establish a law of the iterated logarithm.

Some generalized limit theorems concerning delayed sums of random sequence

For a sequence of arbitrarily dependent m-valued random variables (Xn)n∈N, the generalized strong limit theorem of the delayed average is investigated. In our proof, we improved the method proposed by Liu[6]. As an application, we also studied some limit properties of delayed average for inhomogeneous Markov chains.

Limit theorems for delayed sums of random sequence

For a sequence of arbitrarily dependent random variables (X n )n∈Nand Borel sets (B n )n∈N, on real line the strong limit theorems, represented by inequalities, i.e. the strong deviation theorems of the delayed average are investigated by using the notion of asymptotic delayed log-likelihood ratio. The results obtained popularizes the methods proposed by Liu. Mathematics Subject Classification 2000: Primary, 60F15.

Counting Simsun Permutations by Descents

We count in the present work simsun permutations of length n by their number of descents. Properties studied include the recurrence relation and real-rootedness of the generating function of the number of n-simsun permutations with k descents. By means of generating function arguments, we show that the descent number is equidistributed over n-simsun permutations and n-Andre permutations. We also compute the mean and variance of the random variable Xn taking values the descent number of random n-simsun permutations, and deduce that the distribution of descents over random simsun permutations of length n satisfies a central and a local limit theorem as n → + ∞.

Skorohod representation theorem via disintegrations

Let (µn: n ≥ 0) be Borel probabilities on a metric space S such that µn → µ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn ∼ µn for all n and Xn → X0 in probability. By Skorohod’s theorem, Skorohod representation holds (with Xn → X0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod representation holds if W(µn, µ0) → 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.

Extreme values for Benedicks–Carleson quadratic maps

AbstractWe consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).

Limiting theorems for the nodes in binary search trees

We consider three random variables X n , Y n and Z n , which represent the numbers of the nodes with 0, 1, and 2 children, in the binary search trees of size n. The expectation and variance of the three above random variables are got, and it is also shown that X n , Y n and Z n are all asymptotically normal as n → ∞ by applying the contraction method.

A generalized semi-Pareto minification process

In this paper a generalization of the semi-Pareto autoregressive minification process of the first order is given. The necessary and sufficient condition for stationarity of the process is determined. It is shown that the process is ergodic and uniformly mixing. The joint survival function and the joint density function of the random variables X n+h and X n are determined. The extremes of the random variables X 1, X 2, ..., X n and the geometric extremes of random variables X 1, X 2, ..., X N are derived and their asymptotic distributions are discussed. The estimation of the parameters is discussed and some numerical results are given.