Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Empirical processes of multidimensional systems with multiple mixing properties

We establish a multivariate empirical process central limit theorem for stationary R^d-valued stochastic processes (X_i)_i≥1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling et al. (2009) [9] in the univariate case. As an important technical ingredient, we prove a 2pth moment bound for partial sums in multiple mixing systems.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes for which there is a finite K () such that the conditional distribution of X₁ given the entire past is equal to the conditional distribution of X₁ given only . The least such value of K is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of K, both in the backward sense that we have just described and in the forward sense, where one observes successive values of {Xn} for n?0 and asks for the least value K such that the conditional distribution of Xn+1 given is the same as the conditional distribution of Xn+1 given . We allow for finite or countably infinite alphabet size.     

On the prediction of vector-valued random fields and the spectral distribution of their evanescent component

Let (X_m,n)_(m,n)∈Z² be a C^p-valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z². In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.     

Multivariate generalized Ornstein–Uhlenbeck processes

De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes _(Vt)t≥0${\left(V_t\right)}_{t\ge 0}$ which are basically characterized by the fact that for each h>0$h>0$ the equidistantly sampled process _(Vnh)n∈N0${\left(V_{nh}\right)}_{n\in {\mathbb{N}}_0}$ satisfies the random recurrence equation _Vnh=_{A(n−1)h,nhV(n−1)h}+_B(n−1)h,nh$V_{nh}=A_{(n-1)h,nh}{V}_{(n-1)h}+B_{(n-1)h,nh}$, n∈N$n\in \mathbb{N}$, where _{(A(n−1)h,nh,B(n−1)h,nh)n∈N}${(A_{(n-1)h,nh},B_{(n-1)h,nh})}_{n\in \mathbb{N}}$ is an i.i.d. sequence with positive _A0,h$A_{0,h}$ for each h>0$h>0$. We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Lévy process (X,Y)$(X,Y)$ in R^m×m×R^m${\mathbb{R}}^{m\times m}\times {\mathbb{R}}^m$. The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.     

Scaling limits of Gaussian vector fields

For Gaussian vector fields {X(t) ∈ Rⁿ:t ∈ R^d} we describe the covariance functions of all scaling limits Y(t) = $\text{L}$lim_α↓0 B^?1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions $\text{r(t, s) = EY(t) Y}{}^1\text{(s)}$ are found to be homogeneous in the sense that for some matrix L and each α > 0, $\text{(}{}^1\text{) r(αt, αs) = α}{}^{\mathrm{L1}}\text{r(t, s) α}{}^{\mathrm{L}}$. Processes with stationary increments satisfying (¹) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.     

Potential analysis for positive recurrent Markov chains with asymptotically zero drift: Power-type asymptotics

We consider a positive recurrent Markov chain on R⁺${\mathbb{R}}^{+}$ with asymptotically zero drift which behaves like −_c1/x$-c_1/x$ at infinity; this model was first considered by Lamperti. We are interested in tail asymptotics for the stationary measure. Our analysis is based on construction of a harmonic function which turns out to be regularly varying at infinity. This harmonic function allows us to perform non-exponential change of measure. Under this new measure Markov chain is transient with drift like _c2/x$c_2/x$ at infinity and we compute the asymptotics for its Green function. Applying further the inverse transform of measure we deduce a power-like asymptotic behaviour of the stationary tail distribution. Such a heavy-tailed stationary measure happens even if the jumps of the chain are bounded. This model provides an example where possibly bounded input distributions produce non-exponential output.     

The oscillating random walk

{Y_n;n=0, 1, …} denotes a stationary Markov chain taking values in R^d. As long as the process stays on the same side of a fixed hyperplane E⁰, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′_n = |Y′_n?1?Z_n| (with the Z_n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y_n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.     

Stopping for two-dimensional stochastic processes

The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R²₊. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.     

Parameter estimation and asymptotic stability in stochastic filtering

In this paper, we study the problem of estimating a Markov chain X$X$ (signal) from its noisy partial information Y$Y$, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{_Xn∣_Yn,…,_Y1}$\mathbb{P}\{X_n\mid Y_n,\dots ,Y_1\}$, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.     

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in R^d${\mathbb{R}}^d$ and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable Z_n,α=Y₁+2^αY₂+?+n^αY_n, with α∈R and Y₁,Y₂,… independent and exponentially distributed random variables with mean one. The distribution function of Z_n,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Z_n,α<x) that remains valid inside (α≥−1/2) and outside (α<−1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α=1, for which we sharpen some of the results in Kingman and Volkov (2003).