Variable-Length g-Measures in Stochastic Models

• Variable-length g-measures are probability measures on bi-infinite sequences whose transition probabilities depend on a variable-length, measurable context function, extending classical Markov chains.
• They employ context tree structures and renewal properties to rigorously analyze ergodicity, mixing rates, and conditions for existence and uniqueness of the underlying stochastic process.
• Connections to Gibbsian theory and linear algebraic frameworks underscore their potential in modeling processes with variable memory depths and non-uniform continuity properties.

A variable-length $g$-measure is a probability measure on bi-infinite sequences over a finite alphabet whose specification (the probability of observing a symbol at time $0$ given the infinite past) depends on a context whose length is itself a measurable function of the past. These measures generalize classical finite-state Markov chains and chains with infinite memory, providing a rigorous framework for modeling processes in which memory effects are modulated by recent patterns in the observed sequence. The theory of variable-length $g$-measures is closely connected with context tree models, renewal structures, and stationary stochastic processes with variable memory depth. This article reviews fundamental definitions, structural properties, existence and uniqueness criteria, ergodicity, mixing rates, and their interplay with Gibbsian specifications, referencing the key developments found in (Ferreira et al., 2019, Fernández et al., 2011), and (Cénac et al., 2020).

1. Foundations: $g$-Functions and Compatibility

Let $A$ denote a finite alphabet, and define $\mathcal{X} = A^{\mathbb{Z}}$ as the space of bi-infinite sequences. A $g$-function is a measurable kernel $g: A^{-\mathbb{N}^*}\to[0,1]$ assigning to each semi-infinite past $\underline{x}=x_{-\infty}^{-1}$ a transition probability $g(\underline{x}a)$ for $0$0, normalized so that $0$1. A stationary measure $0$2 on $0$3 is said to be compatible with $0$4 (i.e., is a $0$5-measure) if for $0$6-almost every $0$7, the conditional probability of $0$8 given $0$9 equals $g$0 (Ferreira et al., 2019).

Variable-length $g$1-measures correspond to the case where there exists a measurable context function $g$2, known as the continuity radius, such that $g$3 depends only on the past block $g$4 for almost every $g$5. The set of all such minimal-length contexts produces a context tree, and the collection of context-dependent transition kernels $g$6 (for any $g$7 ending with $g$8) defines the model (Ferreira et al., 2019, Cénac et al., 2020).

2. Context Trees and Renewal Structures

A variable-length memory chain (VLMC) is determined by a context tree $g$9 (a rooted, possibly infinite, saturated tree with context set $g$0) such that each context $g$1 is associated with a probability distribution $g$2 on $g$3. The Markovian process on contexts is then defined via the kernel

$g$4

where $g$5 represents the infinite suffix process and $g$6 denotes the longest context prefix (Cénac et al., 2020). The key combinatorial structure is the longest internal strict suffix (LIS) decomposition, which organizes context transitions and underpins renewal-based representations. In stable context trees (closed under left-shifts), VLMCs admit a decomposition into renewal blocks defined by changes in the LIS, inducing a semi-Markov structure on context labels (Cénac et al., 2020).

The renewal property arises because after a renewal (e.g., a specific finite context), the process "forgets" the details of the further past, akin to regeneration points in classical probabilistic renewal processes. This property has significant implications for perfect simulation and mixing properties (Fernández et al., 2011).

3. Existence and Uniqueness Criteria

The central existence theorem for $g$7-measures establishes that, for any $g$8-function, there exists at least one stationary measure compatible with $g$9 provided the set of discontinuity points of $A$0 has measure zero under some stationary candidate measure constructed from Cesàro averages (Ferreira et al., 2019). Specifically, starting from any past $A$1, one constructs the measure $A$2 via the Ionescu–Tulcea extension and takes Cesàro averages of its shift iterates. If some limit point of these averages assigns zero measure to the set of discontinuities $A$3, then this limit is a stationary $A$4-measure.

For VLMCs (i.e., where the context length is typically finite almost surely), explicit criteria are available:

• Existence holds if the candidate measure almost surely sees finite context lengths: $A$5 (Ferreira et al., 2019);
• Uniqueness is guaranteed if $A$6 is uniformly positive (infimum strictly positive) and the mean context length $A$7 is finite. This follows from the Johansson–Öberg criterion, where the sum of squared variations of $A$8 is controlled by the context-length tail.

In the context tree formalism, existence and uniqueness are further characterized via the convergence of cascade series and the recurrence properties of the associated $A$9-matrix (built from context transitions and LIS decompositions). In stable VLMCs, these criteria reduce to positive recurrence and convergent cascades, with unique stationary measures arising when the mean sojourn time in each alpha-LIS context is finite (Cénac et al., 2020).

4. Ergodic and Mixing Properties

VLMCs with uniformly positive and locally continuous $\mathcal{X} = A^{\mathbb{Z}}$0-functions exhibit strong ergodic properties, including $\mathcal{X} = A^{\mathbb{Z}}$1-mixing (weak Bernoulli) behavior. The absolute regularity coefficients $\mathcal{X} = A^{\mathbb{Z}}$2 decay to zero under conditions where the probability that a context relevant for a block of length $\mathcal{X} = A^{\mathbb{Z}}$3 exceeds length $\mathcal{X} = A^{\mathbb{Z}}$4 vanishes, uniformly over the process. If the sum over $\mathcal{X} = A^{\mathbb{Z}}$5 of these tail probabilities is finite, the unique stationary measure is weak Bernoulli (Ferreira et al., 2019).

Furthermore, explicit mixing rates and convergence in distribution can be deduced in the case of finite or countable context trees, using renewal-theoretic arguments on the induced Markovian or semi-Markovian process on contexts (Cénac et al., 2020). This provides a direct link between the growth rate of the context tree, tail of the context length distribution, and ergodic properties of the process.

5. Connections to Gibbsian Theory

Despite their regularity and well-posed renewal structure, not all regular (continuous, non-null) $\mathcal{X} = A^{\mathbb{Z}}$6-measures associated with variable-length memory chains are Gibbsian in the sense of possessing two-sided quasilocal specifications. Explicit constructions exist where the one-sided continuity (in the past) fails to extend to two-sided (past-and-future) continuity at specific configurations, such as the all-zero past and future. For instance, specific oscillatory constructions on the sequence of transition probabilities can lead to conditional probabilities (given left and right infinite blocks) that do not converge, yielding essential discontinuities and violating the quasilocality criterion (Fernández et al., 2011).

This demonstrates that the class of variable-length $\mathcal{X} = A^{\mathbb{Z}}$7-measures properly contains the set of one-dimensional Gibbs measures, with implications for the transfer of results between chains with complete connections and Gibbs fields (e.g., large deviations and cluster expansion arguments must be checked for quasilocality requirements).

6. Combinatorial and Linear Algebraic Frameworks

The structure of general VLMCs admits a reduction to linear algebraic problems in countable (or finite) dimensions. The set of stationary VLMC probabilities corresponds to nonnegative left-fixed vectors of the LIS (alpha-LIS) $\mathcal{X} = A^{\mathbb{Z}}$8-matrix, subject to normalizing conditions involving cascade series. For stable context trees, this simplifies significantly, with irreducibility and aperiodicity of the context chain (on finitely many contexts) leading to classical results: unique ergodicity and convergence in distribution (Cénac et al., 2020).

VLMCs can encode a broad class of semi-Markov processes. Any finite-state semi-Markov chain can be represented as the initial-letter process of a VLMC on a comb-shaped context tree, with VLMC ergodic properties recovering the standard necessary and sufficient conditions for existence of a unique limiting distribution (finite mean sojourn time per state).

7. Notions of Discontinuity and Random Contexts

Points of discontinuity in the $\mathcal{X} = A^{\mathbb{Z}}$9-function may be essential or non-essential under a stationary $g$0-measure. If discontinuities are non-essential (i.e., can be removed by modifications on a null set), the measure admits a random context representation, reducing the $g$1-function almost everywhere to a convex mixture of finite-step kernels, paralleling Kalikow's theorem for the continuous case. Explicit examples demonstrate the existence of measures compatible with everywhere discontinuous $g$2-functions, as well as constructions where the discontinuity is essential and cannot be eliminated almost surely (via compositions of i.i.d. product measures), affirming the generality of the existence results and the necessity to consider local, measure-theoretic, rather than uniform continuity assumptions (Ferreira et al., 2019).

---

Variable-length $g$3-measures and their associated context tree structures provide a robust and general framework for modeling and analyzing stochastic processes with memory that depends intricately on recent history. The interplay between combinatorial, renewal-theoretic, and measure-theoretic approaches yields comprehensive criteria for existence, uniqueness, mixing, and ergodicity. However, their independence from Gibbsian quasilocality highlights essential differences between one-sided and two-sided specifications in the theory of stochastic processes and statistical mechanics.