Stationary systems of Gaussian processes

We describe all countable particle systems on $\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process $\xi$. We classify all pairs $(\mathfrak{m},\xi)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\mathfrak{m}$ is arbitrary, whereas the process $\xi$ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and $\xi$ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\mathfrak{m}$ has a density of the form $\alpha e^{-\lambda x}$, where $\alpha >0$, $\lambda\in\mathbb{R}$, whereas the process $\xi$ is of the form $\xi(t)=W(t)-\lambda\sigma ^2(t)/2+c$, where $W$ is a zero-mean Gaussian process with stationary increments, $\sigma ^2(t)=\operatorname {Var}W(t)$, and $c\in\mathbb{R}$.Comment: Published in at http://dx.doi.org/10.1214/10-AAP686 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Distribution of levels in high-dimensional random landscapes

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to $\infty$. The random fields considered include costs of assignments, weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington--Kirkpatrick and Edwards--Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.Comment: Published in at http://dx.doi.org/10.1214/11-AAP772 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Functional limit theorems for sums of independent geometric L\'{e}vy processes

Let $\xi_i$, $i\in \mathbb {N}$, be independent copies of a L\'{e}vy process $\{\xi(t),t\geq0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process $$Z_N(t)=\sum_{i=1}^N\mathrm{e}^{\xi_i(s_N+t)}$$ as $N\to\infty$, where $s_N$ is a non-negative sequence converging to $+\infty$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\liminf_{N\to\infty}\log N/s_N>\lambda_2$ for some critical value $\lambda_2>0$, then the limit is an Ornstein--Uhlenbeck process. However, if $\lambda:=\lim_{N\to\infty}\log N/s_N\in(0,\lambda_2)$, then the limit is a certain completely asymmetric $\alpha$-stable process $\mathbb {Y}_{\alpha ;\xi}$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ299 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Extremes of Independent Gaussian Processes

For every $n\in\N$, let $X_{1n},..., X_{nn}$ be independent copies of a zero-mean Gaussian process $X_n=\{X_n(t), t\in T\}$. We describe all processes which can be obtained as limits, as $n\to\infty$, of the process $a_n(M_n-b_n)$, where $M_n(t)=\max_{i=1,...,n} X_{in}(t)$ and $a_n, b_n$ are normalizing constants. We also provide an analogous characterization for the limits of the process $a_nL_n$, where $L_n(t)=\min_{i=1,...,n} |X_{in}(t)|$.Comment: 19 page

Extremes of the standardized Gaussian noise

Let $\{\xi_n, n\in\Z^d\}$ be a $d$-dimensional array of i.i.d. Gaussian random variables and define \SSS(A)=\sum_{n\in A} \xi_n, where $A$ is a finite subset of $\Z^d$. We prove that the appropriately normalized maximum of \SSS(A)/\sqrt{|A|}, where $A$ ranges over all discrete cubes or rectangles contained in $\{1,\ldots,n\}^d$, converges in the weak sense to the Gumbel extreme-value distribution as $n\to\infty$. We also prove continuous-time counterparts of these results.Comment: 18 page