Max-semistable extremal behavior of AR(1)-processes connected with Bernoulli convolutions

Abstract: We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.

• The paper extends geometric partial attraction and merge theorems for maxima of Bernoulli-convolution AR(1) processes to every β∈(0,1), despite unresolved singularity and absolute-continuity properties.
• It represents the stationary distribution through a log-periodic function on a key subinterval and uses extension and reflection methods to characterize the full distribution, while showing that periodicity may fail globally.
• The dependent AR(1) process converges to a Weibull-type max-semistable law with extremal index p, revealing clustering of extremes and a loss of effective independent observations that does not depend on β.

Setting and motivation

The paper studies the simple autoregressive process $X_{k+1}=\beta X_k+\varepsilon_{k+1}$, where the innovations $\varepsilon_k$ take the values $0$ and $1-\beta$ with probabilities $p$ and $q=1-p$, for $\beta\in(0,1)$. The stationary solution $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$ is, up to an affine transformation, a Bernoulli convolution $\sum_{j\ge0}\beta^j Z_j$ with $\mathbb{P}(Z_j=-1)=p$, $\varepsilon_k$0. Prior work [(2608.14155)'s predecessor, Sterk (2025)] established max-semistable extremal behavior for $\varepsilon_k$1, where the support is a Cantor-like set. The contribution here is to extend this to the full range $\varepsilon_k$2, where the stationary distribution may be absolutely continuous or singular depending on delicate arithmetic properties of $\varepsilon_k$3 (Pisot and Garsia numbers, Solomyak's almost-sure absolute continuity result). The paper deliberately does not resolve the singularity/absolute-continuity dichotomy; instead it derives a structural representation of the distribution function that suffices for extreme value theory regardless of that dichotomy.

Structural representation of the stationary distribution

The stationary distribution function $\varepsilon_k$4 satisfies the refinement equation

$\varepsilon_k$5

with support in $\varepsilon_k$6. A first result extends uniqueness of the continuous solution of this functional equation from $\varepsilon_k$7 to all $\varepsilon_k$8, via a contraction argument on the sup-norm that works for the entire parameter range. A symmetry relation $\varepsilon_k$9 is also recorded.

The key structural result is a partial extension of the log-periodic representation known for $0$0: for $0$1 one has

$0$2

with $0$3 continuous and $0$4-periodic. Crucially, the representation holds only on the left part of the support. Numerical approximations for the golden-ratio parameter indicate that the periodicity of $0$5 genuinely breaks down beyond $0$6, so the full analogue of the $0$7 result is false in general. The paper then offers two devices to characterize $0$8 on the whole interval $0$9 in terms of $1-\beta$0: an extension method, which iteratively applies the functional equation outward from $1-\beta$1 and whose critical steps occur exactly at Pisot numbers (e.g., $1-\beta$2 and the plastic number $1-\beta$3), and a reflection method based on the symmetry relation, which closes the gap entirely when $1-\beta$4. These methods yield structural insight but, as the authors concede, do not produce explicit closed forms for $1-\beta$5 or $1-\beta$6.

Two worked examples illustrate when the periodic function degenerates to a constant, which would yield a max-stable (rather than merely max-semistable) limit. For $1-\beta$7, $1-\beta$8 (a Garsia parameter), the extension method rigorously reconstructs the known piecewise-quadratic distribution with a tent-like density, confirming a constant $1-\beta$9 and hence a max-stable limit. For $p$0 with $p$1 the extension is carried out for $p$2, but the authors explicitly state it remains unclear whether the constant-$p$3 ansatz actually satisfies the functional equation; only numerical evidence is offered.

Domain of geometric partial attraction

Since extremes of an i.i.d. sample from $p$4 are governed by the behavior of $p$5 near $p$6, and by the symmetry relation near $p$7, the log-periodic representation of $p$8 suffices. The main i.i.d. result states that with $p$9, $q=1-p$0, and the geometrically growing subsequence $q=1-p$1,

$q=1-p$2

a max-semistable law of Weibull type with parameters $q=1-p$3 and $q=1-p$4. The proof is a direct computation exploiting the periodicity of $q=1-p$5, avoiding general domain-of-attraction criteria. The result thus extends the geometric-partial-attraction property from $q=1-p$6 to the full range $q=1-p$7. The authors note the theorem does not exclude a constant $q=1-p$8 (max-stable limit), and cite numerical evidence in Solomyak's notes for the case $q=1-p$9.

Merge theorems

Convergence along the single subsequence $\beta\in(0,1)$0 is refined into a merge theorem, valid for all $\beta\in(0,1)$1. Writing $\beta\in(0,1)$2 with $\beta\in(0,1)$3 and using the modified normalization $\beta\in(0,1)$4, the i.i.d. merge theorem asserts uniform convergence

$\beta\in(0,1)$5

where $\beta\in(0,1)$6 is the max-semistable limit. The proof is a simplification of Megyesi's general argument, made possible by continuity of $\beta\in(0,1)$7, and proceeds by a subsequence argument: along subsequences with $\beta\in(0,1)$8, pointwise convergence to $\beta\in(0,1)$9 holds, uniformity on compacts follows, and a compactness argument on the relatively compact sequence $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$0 closes the proof. A modified normalization $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$1 additionally makes the limit independent of the accumulation point of $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$2, using the periodicity of $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$3.

The corresponding result for the stationary $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$4 process is the paper's main theorem. The proof follows the blocking technique of Chernick and of Glavaš–Mladenović–Samorodnitsky: the index set is partitioned into long blocks separated by vanishing gaps, and the four error terms (boundary loss, gap loss, asymptotic independence across blocks, and endpoint mismatch) are shown to vanish. The technically demanding part is establishing asymptotic independence between disjoint blocks; this is handled through a careful regular-conditional-probability computation exploiting the decomposition $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$5 with $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$6 independent of the past and bounded by $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$7. The outcome is the $X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$8 merge theorem

$X_k=\sum_{j\ge0}\beta^j\varepsilon_{k-j}$9

and, along $\sum_{j\ge0}\beta^j Z_j$0, uniform convergence to $\sum_{j\ge0}\beta^j Z_j$1. Comparing with the i.i.d. case, the process has extremal index $\sum_{j\ge0}\beta^j Z_j$2, interpretable as loss of i.i.d. degrees of freedom, the reciprocal expected cluster size, or the recurrence rate near unstable fixed points of associated dynamical systems. Notably, the extremal index is independent of $\sum_{j\ge0}\beta^j Z_j$3.

Connections with dynamical systems

Two dynamical interpretations of $\sum_{j\ge0}\beta^j Z_j$4 are given. First, the skew-product map $\sum_{j\ge0}\beta^j Z_j$5 on $\sum_{j\ge0}\beta^j Z_j$6, which for $\sum_{j\ge0}\beta^j Z_j$7 is topologically conjugate to the generalized baker's transformation of Alexander–Yorke, preserves the product-type measure built from $\sum_{j\ge0}\beta^j Z_j$8 in the $\sum_{j\ge0}\beta^j Z_j$9-direction and Lebesgue measure in the driving coordinate; invariance is verified directly on generating rectangles via the functional equation. Since the driving map has exponential decay of correlations, the $\mathbb{P}(Z_j=-1)=p$0-projection of $\mathbb{P}(Z_j=-1)=p$1-orbits serves as a deterministic counterpart of the $\mathbb{P}(Z_j=-1)=p$2 process, linking the results to extreme value laws for deterministic dynamical systems. Second, in the random-iteration framework of Mitrea–Sterk, the probability $\mathbb{P}(Z_j=-1)=p$3 that random compositions of two expanding maps diverge satisfies the same functional equation and boundary conditions as $\mathbb{P}(Z_j=-1)=p$4 for $\mathbb{P}(Z_j=-1)=p$5 and $\mathbb{P}(Z_j=-1)=p$6; by the uniqueness lemma the two functions coincide for $\mathbb{P}(Z_j=-1)=p$7. This identification rests on the uniqueness result and therefore extends verbatim to all $\mathbb{P}(Z_j=-1)=p$8.

Limitations and open questions

The paper is explicit about several boundaries of its results. The log-periodic representation is established only on $\mathbb{P}(Z_j=-1)=p$9, and numerical evidence suggests it does not extend further in general; the extension and reflection methods characterize $\varepsilon_k$00 implicitly but do not yield explicit formulas for $\varepsilon_k$01. Whether the max-semistable limit is genuinely non-max-stable for most parameters — equivalently, whether $\varepsilon_k$02 is non-constant — is unresolved; the constant case is proven only for $\varepsilon_k$03, $\varepsilon_k$04, and conjectured numerically for $\varepsilon_k$05, $\varepsilon_k$06, where the functional-equation verification is left incomplete. The question of which $\varepsilon_k$07 yield singular versus absolutely continuous stationary distributions is inherited from the Bernoulli convolution literature and untouched here. Finally, whether the deterministic skew-product $\varepsilon_k$08 admits an extreme value law with the same limit and extremal index $\varepsilon_k$09 is raised by the analogy but not proven.

Conclusion

The paper completes the max-semistable extreme value analysis of the Bernoulli-convolution AR(1) process across the full parameter range $\varepsilon_k$10. Its two central contributions are a partial log-periodic representation of the stationary distribution function that suffices for extremal analysis despite holding only on a subinterval, and merge theorems — for both i.i.d. sequences and the dependent $\varepsilon_k$11 process — establishing uniform convergence of normalized maxima to powers of a Weibull-type max-semistable law with extremal index $\varepsilon_k$12. The results are robust to the singular/absolutely-continuous dichotomy of the underlying Bernoulli convolution, while the identification of the exact limit law (semistable versus stable) remains tied to open arithmetic questions about the parameter $\varepsilon_k$13.