Phantom Distribution Functions in Extremal Analysis

• Phantom distribution functions are limiting laws that characterize the nondegenerate behavior of maxima when classical extreme-value theory fails due to strong dependence.
• They generalize the concept of the extremal index by providing a meaningful asymptotic description in regimes where maxima exhibit slow growth.
• They offer a rigorous framework for analyzing extreme events in stationary processes, Markov chains, and random fields under weak or anisotropic mixing conditions.

A phantom distribution function is a limiting law describing the asymptotics of maxima in dependent sequences or random fields whose extremal behavior deviates from that governed by standard extreme-value theory. In situations where the classical extremal index vanishes, indicating pathologically slow growth of maxima, these functions provide a nontrivial, nondegenerate limiting description of the distribution of partial maxima, fundamentally extending the reach of extreme-value theory. The concept, introduced by O’Brien (1987), plays a central role in the study of maxima of stationary sequences, Markov chains, and higher-dimensional random fields, particularly under weak, long-range, or nonmixing dependence (Jakubowski et al., 2018, &&&1&&&, Jakubowski et al., 2020).

1. Definition and Core Properties

Given a (possibly nonstationary) sequence of real-valued random variables $\{X_n\}$ on a probability space, let $M_n = \max\{X_0, ..., X_{n-1}\}$. A distribution function $G$ on $\mathbb{R}$ is called a phantom distribution function for $\{X_n\}$ if

$$\lim_{n\to\infty} \sup_{x\in\mathbb{R}} \left| \mathbb{P}(M_n \leq x) - [G(x)]^n \right| = 0.$$

$G$ is termed a continuous phantom distribution function if it is continuous on its support (Jakubowski et al., 2018, Doukhan et al., 2015).

This formalism implies that, for large $n$, the distribution of the partial maximum $M_n$ closely follows that of the maximum of $n$ i.i.d. draws from $G$. The behavior is universal: only the tail of $G$ (up to strict tail equivalence) matters. For sequences with dependence structures precluding a nonzero extremal index, phantom distribution functions yield sharp asymptotic descriptions when the i.i.d. approximation based on the marginal $F$ fails.

In higher dimensions (random fields) indexed by $\mathbb{Z}^d$, the notion generalizes: there, $G$ is a phantom distribution function if

$$\sup_{x \in \mathbb{R}} \left| \mathbb{P}(M_{\mathbf{n}} \leq x) - G(x)^{n_1 n_2 \cdots n_d} \right| \to 0$$

as $n_i \to \infty$ for all $i$ (Jakubowski et al., 2020).

2. Connection with the Extremal Index

In classical extreme-value theory, the extremal index $\theta \in [0,1]$ quantifies the effective number of independent extreme events in a dependent stationary sequence. For $\theta > 0$, maxima $M_n$ grow comparably to the i.i.d. case, up to exponential rescaling, and standard limit laws apply: $$n [1 - F(u_n(\tau))] \to \tau, \quad \mathbb{P}(M_n \leq u_n(\tau)) \to \exp(-\theta\tau).$$ For many dependent models—especially those with strong long-range dependence or certain Markov chain structures—$\theta = 0$. In this case, maxima $M_n$ grow much more slowly: for any $u_n$ with $n [1 - F(u_n)] \to \tau$, $\mathbb{P}(M_n \leq u_n) \to 1$, yielding degenerate limits and precluding classical normalization (Jakubowski et al., 2018, Doukhan et al., 2015).

Nevertheless, such models may admit nontrivial phantom distribution functions $G$, such that $[G(x)]^n$ describes the correct asymptotics of $M_n$. In effect, phantom distribution functions generalize the notion of the extremal index regime $\theta=0$ by supplying meaningful non-i.i.d. maximum laws where the usual approach collapses.

A precise relation is formalized as: $$\lim_{x \to F^{* -}} \frac{1 - G(x)}{1 - F(x)} = \theta$$ where $F$ is the marginal distribution, and $G$ is a regular phantom distribution function. This yields $\theta=0$ when $G$ decays far more slowly than $F$, which is characteristic of slow-growth maxima (Doukhan et al., 2015).

3. Existence, Characterization, and Uniqueness Criteria

For a stationary sequence $\{X_n\}$, several equivalent criteria characterize the existence of (continuous) phantom distribution functions:

• There exists a regular phantom distribution function $G$.
• There exists a nondecreasing sequence $\{u_n\}$ and $y \in (0,1)$ such that

$\mathbb{P}(M_n < u_n) \to y$

and a mixing condition holds:

$$\sup_{p, q \geq 1} \left| \mathbb{P}(M_{p+q} < u_n) - \mathbb{P}(M_p < u_n)\mathbb{P}(M_q < u_n)\right| \to 0$$

as $n \to \infty$.

• For every $\beta > 0$, there exists $v_n \uparrow \infty$ such that for all $t \geq 0$,

$$\mathbb{P}(M_{\lfloor nt \rfloor} \leq v_n) \to \exp(-\beta t)$$

and, when these hold for a dense set $D \subset \mathbb{R}_+$ and $\{v_n\}$ nondecreasing, an explicit continuous $G$ can be constructed (Jakubowski et al., 2018, Doukhan et al., 2015).

Uniqueness is up to strict tail-equivalence: any two regular phantom distribution functions for the same sequence are strictly tail-equivalent.

For random fields over $\mathbb{Z}^d$, analogous results hold, with the key property being strong block factorization (Condition $\mathbf{B}_T$) along all monotone curves in the indexing lattice (Jakubowski et al., 2020).

4. Role in Weak and Dependent Structures

Phantom distribution functions are prevalent among stationary sequences and fields that fail strong mixing, or exhibit only weak, e.g., $\alpha$-, $\theta$-, $\eta$-, or $\kappa$-dependence. Major results include:

• Any $\alpha$-mixing stationary sequence with continuous marginals admits a continuous phantom distribution function (Doukhan et al., 2015).
• Sufficient mixing-type covariance constraints—and appropriate truncation arguments—guarantee existence even under various generalized weak dependence structures.
• For random fields, global phantom distribution functions require decorrelation along all growth routes; directional (sectorial) phantom distribution functions allow for block-independence only along chosen sectors or multidimensional "diagonals" (Jakubowski et al., 2020).
• For sequences or fields with discontinuous marginals, existence still follows under additional regularity conditions on the jump sizes and their relative tail decay (Doukhan et al., 2015).

The interpretation is that phantom distribution functions accurately capture the extremal behavior in a broad class of dependent models not encompassed by classical extremal index theory.

5. Markov Chains and Quenched Results

A central development is the identification of phantom distribution limits for positive Harris-recurrent aperiodic Markov chains $\{Y_n\}$ on general state spaces. If $f: S \to \mathbb{R}$ is measurable, $X_n = f(Y_n)$, and under the stationary distribution $\pi$,

$\sup_x |\mathbb{P}_\pi(M_n \leq x) - G(x)^n| \to 0$

for a continuous phantom distribution function $G$, then, importantly, this convergence extends to all initial distributions $\lambda$ with $\lambda(S_0)=0$ for a $\pi$-null set $S_0$. When $\pi \circ f^{-1}$ is continuous and unbounded above, the convergence holds for every starting state. The proof utilizes coupling arguments for convergence of Markov chains and the existence criteria for phantom distributions (Jakubowski et al., 2018).

This "quenched" result implies that, for many ergodic Markov chains—including random-walk Metropolis algorithms with heavy-tailed targets—the slow-growth, extremal-index-zero regime is universally described by a phantom law, independent of initialization.

6. Illustrative Examples and Applications

Canonical examples include:

• The random-walk Metropolis algorithm targeting a heavy-tailed distribution. For subexponential or regularly varying tails, the extremal index is zero, yet a continuous phantom distribution function $G$ accurately describes the maxima. For all $x$,

$$\mathbb{P}_x \left( \max_{1 \leq j \leq n} X_j \leq x \right) \asymp G(x)^n, \quad n \to \infty$$

(Jakubowski et al., 2018, Doukhan et al., 2015).

• The reflected random walk (Lindley's process) in heavy-tailed regimes demonstrates the existence of phantom distributions even when the stationary tail is heavier than that of the increments, and the extremal index vanishes (Doukhan et al., 2015).
• Non-ergodic mixtures (e.g., exchangeable processes with artificial blockwise jump laws) that, despite lacking ergodicity, admit continuous phantom distributions in the described limit-theoretic sense (Doukhan et al., 2015).
• Random fields: certain stationary Gaussian fields exhibit only sectorial phantom distribution functions (approximation holds along the diagonal), but not globally, exemplifying the need for directional generalizations in high dimensions (Jakubowski et al., 2020).

7. Sectorial and Directional Phantom Distribution Functions

For stationary random fields, it is often impossible to achieve uniform phantom approximation over all growth directions. The sectorial (or directional) phantom distribution function is defined by restricting attention to all monotone curves within a sector about a preferred direction. For the diagonal $\boldsymbol{\Delta}(n) = (n, ..., n)$, if

$$\sup_{x\in\mathbb{R}}\left|\mathbb{P}(M_{\boldsymbol{\varphi}(n)} \leq x) - G(x)^{\boldsymbol{\varphi}(n)^*}\right| \to 0$$

uniformly for all monotone $\boldsymbol{\varphi}$ staying within powers of $n$ of the diagonal, $G$ is a sectorial phantom distribution function (Jakubowski et al., 2020).

In such fields, the sectorial extremal index may exist even if no global extremal index does, capturing anisotropic or directionally dependent structures. Limit theorems along these sectors yield a full family of convergence results, formalizing the manner in which maxima "along a direction" display nondegenerate phantom limits.

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References

• "Quenched phantom distribution functions for Markov chains" (Jakubowski et al., 2018)
• "Phantom distribution functions for some stationary sequences" (Doukhan et al., 2015)
• "Directional phantom distribution functions for stationary random fields" (Jakubowski et al., 2020)