The k-record processes are i.i.d.

Let X ₁, X ₂, ... be i.i.d. positive random variables, and let _n be the initial rank of X _n (that is, the rank of X _n among X ₁, ..., X _n). Those observations whose initial rank is k are collected into a point process N ^k on ⁺, called the k-record process. The fact that {itN^k; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N ¹, ..., N ^k, the lifetimes in rank k of all observations of initial rank at most k are independent geometric random variables.These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a time-space Poisson process. Initially, we think of this Poisson process as having values in ⁺, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes.     

A two-sided estimate in the hsu-robbins-erdös law of large numbers for i.i.d. random variables sequence

The Paley-type inequalities for the complete convergence and lower and upper bounds in the Baum-Katz law of large numbers for i.i.d. random variables sequence are obtained. The results obtained generalize the result of Pruss [8].     

Tightness of products of i.i.d. random matrices

Summary In this paper we present a necessary and sufficient condition for tightness of products of i.i.d. finite dimensional random nonnegative matrices. We give an example illustrating the use of our theorem and treat completely the case of 2×2 matrices. We also describe stationary solutions of the linear equationy _n=X_ny_n–1, n>0, in (R ^d )⁺, whereX ₁,X ₂,... are i.i.d.d×d nonnegative matrices.     

Complete moment convergence for i.i.d. random variables

In the paper, the complete moment convergence is obtained for i.i.d. random variables such that all moments exist, but the moment generating function does not exist. The main results extend the related known works due to Gut and Stadtmüller.     

Random coverings of the circle with i.i.d. centers

Consider the random intervals In(ω):=(ωn-ln/2,ωn+ln/2)(mod 1) with their centers ωn being i.i.d.but not necessary uniformly distributed on the circle T = R /Z and with their lengths decreasing to zero.Using the dimension theory in dynamical systems,we give conditions on which the circle is finitely or infinitely often covered by intervals In(ω)}n≥1.     

On the Number of i.i.d. Samples Required to Observe All of the Balls in an Urn

Suppose an urn contains m distinct balls, numbered 1,...,m, and let τ denote the number of i.i.d. samples required to observe all of the balls in the urn. We generalize the partial fraction expansion type arguments used by Pólya (Z Angew Math Mech 10:96–97, 1930) for approximating \mathbbE(t)\mathbb{E}(\tau) in the case of fixed sample sizes to obtain an approximation of \mathbbE(t)\mathbb{E}(\tau) when the sample sizes are i.i.d. random variables. The approximation agrees with that of Sellke (Ann Appl Probab 5(1):294–309, 1995), who made use of Wald’s equation and a Markov chain coupling argument. We also derive a new approximation of \mathbbV(t)\mathbb{V}(\tau), provide an (improved) bound on the error in these approximations, derive a recurrence for \mathbbE(t)\mathbb{E}(\tau), give a new large deviation type result for tail probabilities, and look at some special cases.     

Asymptotic expansions for potential functions of i.i.d. random fields

Summary For sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two. Under standard conditions, formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters. If this convergence fails, the distribution of the sum can be approximated by a mixture of lattice distributions. The result applies to m-dependent random fields generated by independent random variables.     

Trimmed sums of i.i.d. Banach space valued random variables

Limit laws for trimmed sums of triangular arrays of i.i.d. Banach space valued random variables are studied. It is shown that if the array belongs to the domain of attraction of an infinitely divisible law without Gaussian component on a separable Banach space of type 2, then the trimmed sum converges weakly to a nondegenerate Banach space valued random variable.     

Almost sure rates of mixing for i.i.d. unimodal maps

It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young [8], and Baladi and Viana [4] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the “averaged statistics”. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [1], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.     

Large deviations for sums of i.i.d. random compact sets

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

     

Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

We study the upper-lower class behavior of weighted sums ∑ _k=1 ⁿ a _k X _k , where X _k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X _j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a _n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a _n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.     

A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a _in} are uniformly bounded,E|X|^1/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.     

Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed driving noise. For general ARMA(p, q) equations these conditions are expressed in terms of the coefficient polynomials of the defining equations and moments of the driving noise sequence, while for p =?1 an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices.     

A note on occurrence of gapped patterns in i.i.d. sequences

A new martingale technique is developed to find formulas for the expected value and generating function of the waiting time until one observes a gapped pattern (or a structured motif) in an i.i.d. sequence of random letters from a finite alphabet.     

Limit laws for power sums and norms of i.i.d. samples

We study the limit behavior of power sums and norms of i.i.d. positive samples from the max domain of attraction of an extreme value distribution. To this end, we combine limit theorems for sums and for maxima and use a link between extreme value theory and the Lévy measures of certain infinitely divisible laws, which are limit distributions of power sums. In connection with the von Mises representation of the Gumbel max domain of attraction, this new approach allows us to extend the limit results for power sums found in Ben Arous et al. (Probab Theory Relat Fields 132:579–612, 2005) and Bogachev (J Theor Probab 19:849–873, 2006). Furthermore, our findings shed a new light on the results of Schlather (Ann Probab 29:862–881, 2001) and treat the Gumbel case which is missing there.     

Asymptotics for self-normalized random products of sums of i.i.d. random variables

Let be a sequence of independent and identically distributed positive random variables, which is in the domain of attraction of the normal law, and tn be a positive, integer random variable. Denote , , where denotes the sample mean. Then we show that the self-normalized random product of the partial sums, , is still asymptotically lognormal under a suitable condition about tn.     

Lim sup behavior of sums of geometrically weighted i.i.d. random variables

Geometrically weighted i.i.d. random variables {Y_n} which are bounded above are shown to exhibit iterated logarithm type behavior. Specifically, if b > 1 and if the lower tail of the distribution of Y₁ approaches 0 fast enough, then lim sup_n→∞(b?1) Σⁿ_j=1b¹Y_j?bⁿ⁺¹=L, almost certainly, where L is the essential supremum of Y₁.