Local Maxima of Discrete Gaussian Processes: Exact Distributions and Infill Limits

Introduction

Local maxima provide natural candidates for detecting signals in temporal and spatial data. Applications include neuroimaging, genomics, change point analysis, and environmental signal detection. Selecting a point because it is a local maximum changes its null height distribution: a marginally Gaussian process need not have Gaussian heights at its selected peaks. Conditional peak height distributions therefore provide a building block for inference at local maxima [5, 6, 9, 18].

For smooth Gaussian processes and fields, expected numbers and height distributions of local maxima are studied through Kac–Rice formulas [1, 2, 10, 17]. Explicit peak height distributions and their dependence on covariance structure have been developed in [4, 7, 8]. There is an essential distinction between continuous and discrete parameter spaces. Under the usual nondegeneracy and smoothness conditions, a fixed deterministic point of a continuous process is a local maximum with probability zero. The height distribution of a continuous peak is then defined through local Palm sampling and calculated by Kac–Rice. On a discrete grid, the event that a specified site exceeds its two neighbors has positive probability under the finite-dimensional nondegeneracy conditions, so its height has an ordinary conditional law.

The discrete law is directly relevant when data are recorded at finitely many locations and peaks are identified by comparisons between neighboring observations. A continuous approximation may be accurate for a sufficiently fine grid and a sufficiently smooth process, but its accuracy depends on both the covariance and the mesh. Discrete local maximum methods for bounding excursion probabilities were developed by [20, 21]. In particular, Taylor et al. [20] calculate the conditional probability that neighboring observations fall below a given central height and integrate this probability against the Gaussian density. In one dimension, that calculation is equivalent to the stationary joint height and peak probability used here. Lin et al. [13] explicitly normalize such probabilities to obtain finite-lattice peak height laws and develop a Monte Carlo method for general stationary neighborhood covariance matrices.

The present paper complements these finite-lattice calculations with nonstationary formulas and rigorous infill limits. We establish convergence to a curvature-weighted density under fourth-order covariance smoothness, give a necessary and sufficient nonstationary criterion for a Gaussian height limit, and classify stationary limits through the fourth spectral moment. Remark 4.2 organizes the resulting height laws in the continuous family $\{h_{\eta}^{\mathrm{SM}}: 0 \le\eta\le1\}$, whose endpoints are Gaussian and Rayleigh. The finite-dimensional Gaussian conditioning calculation is an established ingredient; our principal results concern how discrete peak selection and covariance regularity determine the limiting law and the accuracy of its finite-grid approximation.

Let $X = \{X(t): t \in I\}$ be a centered, unit-variance, real-valued Gaussian process on an interval $I \subset\mathbb{R}$, with covariance

In the stationary case, we write $C(s,t) = r(s-t)$. Under stationarity, since the quantities below do not depend on location, we omit their subscript $t$; for the spectral results, $X$ is understood to be defined on $\mathbb{R}$ and observed only on $I$.

For a mesh size $d > 0$ and a site whose two neighbors lie in $I$, write the event of observing a local maximum at $t$ as

We study the sitewise peak probability $p_{t,d} = \mathbb{P}(\mathcal{A}_{t,d})$ and the conditional peak height density

The main contributions are as follows.

1. We express $p_{t,d}$ and $h_{t,d}$ using a bivariate Gaussian orthant probability and a bivariate Gaussian distribution function. The formulas require only the covariance matrix of $\{X(t-d), X(t), X(t+d)\}$ and no sample path differentiability. Summing the sitewise peak probabilities and integrating the joint height densities gives exact expected counts of all grid peaks and of grid peaks above a threshold.

2. When $C$ has four derivatives near the diagonal, we prove that $p_{t,d}/d$ converges to a curvature-weighted intensity and $h_{t,d}$ converges to $h^{\mathrm{SM}}_{\eta_t}$, with $\eta_t \in(0,1]$; see Theorem 3.2. The height law depends on the single parameter $\eta_t$, the magnitude of the conditional height-curvature correlation. The endpoint $\eta_t = 1$ has zero conditional curvature variance and a Rayleigh height law. Under additional sample path and Kac–Rice regularity assumptions, these limits describe the intensity and Palm height distribution of continuous local maxima.

3. For general nonstationary covariance, let $v^{\pm}_{t,d} = 1 - C(t,t \pm d)$ and let $q_{t,d}$ be the correlation between $X(t) - X(t-d)$ and $X(t) - X(t+d)$. We prove that

is necessary and sufficient for weak convergence of peak heights to $N(0,1)$, and is equivalent to pointwise and $L^1$ convergence of their densities. The criterion allows the peak probability to vanish and does not require convergence of $q_{t,d}$. Explicit finite-grid bounds give uniform sitewise approximations and control relative tail errors over a specified threshold range. In the stationary case, $R_d = 4\{1-r(d)\}^2/[4\{1-r(d)\} - \{1-r(2d)\}]$. The criterion includes nondifferentiable processes and processes with one, but not two, mean-square derivatives.

4. For each fixed nonconstant, mean-square continuous stationary process with spectral moments $m_j$, we give a complete classification. If $m_4 < \infty$, the limiting density is $h^{\mathrm{SM}}_{\eta}$ with $\eta= m_2/\sqrt{m_4} \in(0,1]$. If $m_4 = \infty$, the limit is the Gaussian endpoint $h^{\mathrm{SM}}_0 = \phi$, and expected peak counts diverge on every interval of positive length. The Matérn family illustrates this boundary, including its logarithmic cases.

The unification in Remark 4.2 gives a common description of the limiting height laws established in this paper. The smooth density $h^{\mathrm{SM}}_{\eta}$ extends continuously to the Gaussian law as $\eta\downarrow0$ and to the Rayleigh law as $\eta\uparrow1$, with convergence in total variation at both endpoints. In the smooth regime, $\eta$ measures the relation between height and curvature conditional on zero slope; the Gaussian endpoint describes asymptotically height-neutral peak selection. For a fixed stationary process, the spectral classification specifies the member that arises: $\eta= 0$ when $m_4 = \infty$, and $\eta= m_2/\sqrt{m_4} > 0$ when $m_4 < \infty$. The interpolation of the family therefore describes variation across covariance models, while the infill theorems identify the limits under their respective assumptions.

The difference between the smooth and rough regimes is fundamental. For a smooth process, the probability that a specified grid point is a local maximum is of order $d$; summing over approximately $d^{-1}$ sites produces a finite limiting number of continuous local maxima. The comparison event concentrates near zero slope, with a width proportional to negative curvature, and produces the curvature weighting in $h^{\mathrm{SM}}_{\eta t}$ which is generally non-Gaussian. For a rough process, grid peaks become increasingly numerous as the grid is refined. Whether $t$ is a discrete local maximum is determined by the increments from $X(t)$ to its neighboring grid values. As the grid spacing tends to zero, these increments become asymptotically independent of $X(t)$, so the local maximum event carries essentially no information about the height at $t$. This explains why the marginal Gaussian density reappears after conditioning.

The rest of the paper is organized as follows. Section 2 derives the exact finite-mesh distributions and expected count formulas. Section 3 proves the smooth infill limit and its stationary specialization. Section 4 gives the nonstationary Gaussian-limit criterion, the unified family interpretation, and the stationary spectral classification. Section 5 treats common covariance models, with stationary and nonstationary examples and numerical comparisons. Section 6 discusses approximation choices and the scope of the inference implications.

Exact Discrete Distributions

Grid peaks and notation

Let $I = [a,b]$ and, for $d > 0$, define

We restrict attention to grid points whose two neighboring grid points also lie in $I$. Boundary points, which require a separate one-sided definition of a local maximum, are not considered here. Throughout, $X$ is centered and $\operatorname{Var}\{X(t)\} = 1$. Write $\phi$, $\Phi$, and $\Psi= 1 - \Phi$ for the standard Gaussian density, distribution function, and survival function. For $\rho\in[-1,1]$, let $\Phi_2(x,y;\rho)$ be the distribution function of a standard bivariate Gaussian vector with correlation $\rho$. This definition includes the singular cases:

where $z_+ = \max\{z,0\}$.

For $t \in\mathcal{G}^{\circ}_d$, define the event of local maximum at $t$ and the corresponding probability as

Whenever $p_{t,d}>0$, define respectively the cumulative distribution function and the density function of the height of a local maximum at $t$ as

Denote by $\overline{H}_{t,d}(u)=1-H_{t,d}(u)$ the tail of peak height distribution.

The exact nonstationary formula

For compact notation, set

When $c_{\pm}<1$, the two comparison differences $X(t)-X(t-d)$ and $X(t)-X(t+d)$ have positive variances $2(1-c_{\pm})$, so ties between the center and either neighbor have probability zero. Their correlation is

For $c_{\pm}\in(-1,1)$, also let