Finite-dimensional Distributions Research Articles

CRPS M-estimation for max-stable models

Max-stable processes provide canonical models for the dependence of multivariate extremes. Inference with such models has been challenging due to the lack of tractable likelihoods which has motivated use of composite likelihood methods Padoan et al. (J. Amer. Stat. Assoc. 105(489):263–277, 2010). In contrast, the finite dimensional cumulative distribution functions (CDFs) appear natural to work with, and are readily available or can be approximated well. Motivated by this fact, in this work we develop an M-estimation framework for max-stable models based on the continuous ranked probability score (CRPS) of multivariate CDFs. We start by establishing conditions for the consistency and asymptotic normality of the CRPS-based estimators in a general context. We then implement them in the max-stable setting and provide readily computable expressions for their asymptotic covariance matrices. The resulting point and asymptotic confidence interval estimates are illustrated over popular simulated models. They enjoy accurate coverages and offer an alternative to composite likelihood based methods.

An Approximation of Subfractional Brownian Motion

In this article, we obtain an approximation theorem for subfractional Brownian motion with H > 1/2, using martingale differences. The proof involves the tightness and identification of finite dimensional distributions.

Limit theorems for renewal shot noise processes with eventually decreasing response functions

We consider shot noise processes (X(t))t≥0 with deterministic response function h and the shots occurring at the renewal epochs 0=S0<S1<S2⋯ of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut))u≥0 as t→∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut))u≥0 converge weakly as t→∞. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index −β (with 0≤β<1/α or 0≤β≤α, depending on whether ES1<∞ or ES1=∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut))u≥0. The limits are fractionally integrated stable Lévy motions if ES1<∞ and fractionally integrated inverse stable subordinators if ES1=∞.

Large deviations for affine diffusion processes on [formula omitted

This paper proves the large deviation principle for affine diffusion processes with initial values in the interior of the state space R+m×Rn. We approach this problem in two different ways. In the first approach, we first prove the large deviation principle for finite dimensional distributions, and then use it to establish the sample path large deviation principle. For this approach, a more careful examination of the affine transform formula is required. The second approach exploits the exponential martingale method of Donati-Martin et al. for the squares of Ornstein–Uhlenbeck processes. We provide an application to importance sampling of affine diffusion models.

Testing over a continuum of null hypotheses with False Discovery Rate control

We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding $p$-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the $p$-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of $p$-values, or under the assumption that the finite dimensional distributions of the $p$-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.

Calculation of ruin probabilities for a dense class of heavy tailed distributions

In this paper, we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional phase-type distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.

MIMICKING FINITE DIMENSIONAL MARGINALS OF A CONTROLLED DIFFUSION WITH JUMPS

Given a controlled diffusion with jumps, it is shown under some conditions that there exists a Markov controlled diffusion which has the same cost under any criterion which only depends on the one-dimensional distributions. This result is then used to prove that the finite dimensional marginal distribution of a controlled diffusion with jumps at a prescribed set of time instants can also be attained by using a control from a much smaller class of controls called "nearly Markov controls", extending the work of Borkar [4] to the discontinuous setting.

On the uniform convergence of random series in Skorohod space and representations of càdlàg infinitely divisible processes

Let $X_n$ be independent random elements in the Skorohod space $D([0,1];E)$ of c\`{a}dl\`{a}g functions taking values in a separable Banach space $E$. Let $S_n=\sum_{j=1}^nX_j$. We show that if $S_n$ converges in finite dimensional distributions to a c\`{a}dl\`{a}g process, then $S_n+y_n$ converges a.s. pathwise uniformly over $[0,1]$, for some $y_n\in D([0,1];E)$. This result extends the It\^{o}-Nisio theorem to the space $D([0,1];E)$, which is surprisingly lacking in the literature even for $E=R$. The main difficulties of dealing with $D([0,1];E)$ in this context are its nonseparability under the uniform norm and the discontinuity of addition under Skorohod's $J_1$-topology. We use this result to prove the uniform convergence of various series representations of c\`{a}dl\`{a}g infinitely divisible processes. As a consequence, we obtain explicit representations of the jump process, and of related path functionals, in a general non-Markovian setting. Finally, we illustrate our results on an example of stable processes. To this aim we obtain new criteria for such processes to have c\`{a}dl\`{a}g modifications, which may also be of independent interest.

Ergodicity and mixing bounds for the Fisher–Snedecor diffusion

We consider the Fisher-Snedecor diffusion; that is, the Kolmogorov-Pearson diffusion with the Fisher-Snedecor invariant distribution. In the nonstationary setting, we give explicit quantitative rates for the convergence rate of respective finite-dimensional distributions to that of the stationary Fisher-Snedecor diffusion, and for the $\beta$-mixing coefficient of this diffusion. As an application, we prove the law of large numbers and the central limit theorem for additive functionals of the Fisher-Snedecor diffusion and construct $P$-consistent and asymptotically normal estimators for the parameters of this diffusion given its nonstationary observation.

Random A-permutations and Brownian motion

We consider a random permutation τn uniformly distributed over the set of all degree n permutations whose cycle lengths belong to a fixed set A (the so-called A-permutations). Let Xn(t) be the number of cycles of the random permutation τn whose lengths are not greater than nt, t ∈ [0, 1], and \(l(t) = \sum\nolimits_{i \leqslant t,i \in A} {1/i,t > 0} \). In this paper, we show that the finite-dimensional distributions of the random process \(\{ Y_n (t) = (X_n (t) - l(n^t ))/\sqrt {\varrho \ln n} ,t \in [0,1]\} \) converge weakly as n → ∞ to the finite-dimensional distributions of the standard Brownian motion {W(t), t ∈ [0, 1]} in a certain class of sets A of positive asymptotic density ϱ.

Extremal [formula omitted] processes: Elliptical domain of attraction and a spectral representation

The extremal t process was proposed in the literature for modeling spatial extremes within a copula framework based on the extreme value limit of elliptical t distributions (Davison et al. (2012) [5]). A major drawback of this max-stable model was the lack of a spectral representation such that for instance direct simulation was infeasible. The main contribution of this note is to propose such a spectral construction for the extremal t process. Interestingly, the extremal Gaussian process introduced by Schlather (2002) [22] appears as a special case. We further highlight the role of the extremal t process as the maximum attractor for processes with finite-dimensional elliptical distributions. All results naturally also hold within the multivariate domain.

Markov-modulated infinite-server queues with general service times

This paper analyzes several aspects of the Markov-modulated infinite-server queue. In the system considered (i) particles arrive according to a Poisson process with rate $$\lambda _i$$ ? i when an external Markov process (background process) is in state $$i$$ i , (ii) service times are drawn from a distribution with distribution function $$F_i(\cdot )$$ F i ( · ) when the state of the background process (as seen at arrival) is $$i$$ i , (iii) there are infinitely many servers. We start by setting up explicit formulas for the mean and variance of the number of particles in the system at time $$t\ge 0$$ t ? 0 , given the system started empty. The special case of exponential service times is studied in detail, resulting in a recursive scheme to compute the moments of the number of particles at an exponentially distributed time, as well as their steady-state counterparts. Then we consider an asymptotic regime in which the arrival rates are sped up by a factor $$N$$ N , and the transition times by a factor $$N^{1+\varepsilon }$$ N 1 + ? (for some $$\varepsilon >0$$ ? > 0 ). Under this scaling it turns out that the number of customers at time $$t\ge 0$$ t ? 0 obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.

On Finite-Dimensional Distributions of a Random Element Conditioned by the Sum of Random Elements*

A criterion for a random element to be conditioned by a sum of random elements is proved in terms of finitedimensional distributions.

Exchangeable sequences driven by an absolutely continuous random measure

Let $S$ be a Polish space and $(X_{n}:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_{n}(\cdot)=P(X_{n+1}\in\cdot\mid X_{1},\ldots,X_{n})$ be the predictive measure and $\alpha$ a random probability measure on $S$ such that $\alpha_{n}\stackrel{\mathrm{weak}}{\longrightarrow}\alpha$ a.s. Two (related) problems are addressed. One is to give conditions for $\alpha\ll\lambda$ a.s., where $\lambda$ is a (nonrandom) $\sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal{L}(X_{1},\ldots,X_{n})$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alpha_{n}-\alpha\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert\cdot\Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_{n})$ is conditionally identically distributed, in the sense of [Ann. Probab. 32 (2004) 2029–2052].

Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive Levy process with nonzero Levy measure. In this paper, we study the asymptotic behavior of the local time process, in thespatialvariable,oftheseprocesseskilledattwodifferentrandomtimes:eitheratthe timeofthefirstvisitoftheLevyprocessto0,inwhichcaseweproveresultsattheexcur- sion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under gen- eral assumptions, even if the limiting process is not cadlag. Making an assumption on the distribution of the jumps of the compound Poisson processes, we strengthen this to get weak convergence. Our assumption allows for the limiting process to be a stable Levy process with drift. These results have implications on branching processes and in queueing theory, namely, on the scaling limit of binary, homogeneous Crump- Mode-JagersprocessesandonthescalinglimitoftheProcessor-Sharingqueuelength process.

Weak convergence of subordinators to extremal processes

For certain subordinators (Xt)t≥0 it is shown that the process (−tlogXts)s>0 tends to an extremal process (η̂s)s>0 in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that (z∧(−tlogXts))s≥0 converges weakly to (z∧η̂s)s≥0 in D[0,∞), the space of càdlàg functions equipped with Skorohod’s J1 metric.

Asymptotics of Markov Kernels and the Tail Chain

An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

Asymptotics of Markov Kernels and the Tail Chain

An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

Decomposable Statistics and Random Placement of Particles over a Countable Set of Cells*

In the paper, the scheme of independent particle allocation is considered. A connection is demonstrated between finite-dimensional distributions for the process of the number of particles assigned to the k-th cell and the conditional distribution of independent Poisson distributed random variables. Conditions are specified for the convergence of these distributions to finite-dimensional distributions of the Gaussian process. A certain scheme of dependent particle allocation on a countable set of cells is introduced and investigated.

Some new classes of stationary max-stable random fields

We present two new classes of stationary max-stable random fields. For the first class, we use the spectral representation due to Schlather (2002) and assume that the stationary process used in the representation is proportional to a power of a max-stable random field. We derive the finite dimensional distributions, explain the relationship between distributions of both max-stable random fields and give sufficient conditions for the sample paths to be γ-Hölder continuous functions for some γ∈(0,1). For the second class, we consider a multiplicative factor model and a Poisson–Voronoï tessellation of Rd to construct new max-stable random fields. We provide explicit expressions for the pairwise distribution function.