Multiplicative Lindley’s Recursion

• Multiplicative Lindley’s recursion is defined by updating the process state using a random multiplicative factor and an additive term, generalizing the classical Lindley equation.
• The methodology leverages transform techniques and fixed-point arguments to analyze stability and derive explicit expressions for stationary and transient distributions, particularly in Markov-modulated settings.
• Applications include queueing systems, autoregressive time series, Lévy process fluctuations, and arithmetic dynamics, providing explicit formulas for numerical evaluation and modeling complex dependencies.

Multiplicative Lindley’s Recursion refers to a broad class of stochastic recursions that generalize the classical (additive) Lindley equation to settings where the process updates via multiplicative as well as additive or state-dependent terms, possibly under the influence of additional structural dependencies such as Markov modulation. This topic has attracted considerable research interest over the last decade, leading to technical advancements in queueing theory, stochastic autoregressive models, fluctuation theory for Lévy processes, and the analysis of multidimensional or vector-valued recursive systems.

1. Fundamental Formulation and Distinguishing Features

The canonical multiplicative Lindley recursion is given by: $W_{n+1} = [V_n W_n + Y_n(V_n)]^{+}$ where $[x]^{+} = \max\{x,0\}$, $V_n$ is a random multiplicative factor, and $Y_n(V_n)$ is a random additive term which may depend on $V_n$ and, in the Markov-modulated case, on the underlying background chain $\{Z_n\}$.

This recursion generalizes the classical Lindley equation: $W_{n+1} = \max\{0, W_n + (B_n - A_n)\}$ by allowing the previous state to be scaled (and potentially sign-flipped) in every step. The multiplicative term $V_n$ can have a variety of distributions (continuous, discrete, possibly negative), and its statistical properties—mean, variance, contraction—are central to process stability and the existence of stationary distributions (Boxma et al., 2020).

For queueing-theoretic and autoregressive applications, the recursion is frequently embedded into a Markov-modulated framework, capturing complex dependency structures and model heterogeneity (Dimitriou, 2 Jul 2025, Dimitriou, 28 Aug 2025).

2. Stability and Existence of Stationary Laws

Recursion stability is governed by conditions on the multiplicative factor $V_n$ and the nature of the additive noise $Y_n$:

• If $V_n$ is positive and $\mathbb{E} \log|V_n| < 0$, the process contracts on average, guaranteeing that $\{W_n\}$ converges in law to a stationary distribution.
• If negative values are allowed for $V_n$, stability analysis must also consider the drift and regenerative properties of the additive component. Specific criteria include:
  • Cases where $V_n$ is negative but $Y_n$ is sufficiently dissipative.
  • Cases where $V_n \geq 0$ but $P(V_n = 0) > 0$, making the process intermittently regenerative.
• Stationarity fails when the contraction condition is violated ($\mathbb{E}\log|V_n| \geq 0$) or when the noise drives the process upward indefinitely (Boxma et al., 2020).

All current technical analyses leverage transform methods (Laplace–Stieltjes or Fourier) and fixed-point arguments on contractive metric spaces.

3. Explicit Solutions: Transforms and Recursive Forms

A distinctive methodological advance is the derivation of explicit transform representations for transient and stationary distributions:

• The Laplace–Stieltjes transform (LST) of the stationary law often satisfies a functional equation: $$\widetilde{W}(s) = R(s) \widetilde{W}(a s) + \widetilde{N}(s)$$ where $a$ is a contraction, $R(s)$ is a matrix or scalar kernel, and $\widetilde{N}(s)$ collects regeneration and boundary terms (Dimitriou, 28 Aug 2025, Boxma et al., 2020).

For Markov-modulated recursions, the stationary LST becomes vector-valued: $$\widetilde{\mathbf{W}}(s) = R(s) \widetilde{\mathbf{W}}(a s) + \widetilde{\mathbf{N}}(s)$$ with all coefficients computed from the transition matrix and the LSTs of the various random variables conditional on the state.

Recursive computation of densities/moments is possible when increments have tractable forms, e.g., Laplace or exponential. Explicit formulas for densities and exit-time distributions are given in (Lucrezia et al., 2023), showing recursive structures for the atom at zero and piecewise polynomial-exponential densities over stratified domains.

4. Multidimensional, Vector-Valued, and Markov-Modulated Versions

Extension to multidimensional and vector-valued recursions introduces matrix-valued functional equations: $$\mathbf{Z}_{n+1} = [R_n(\mathbf{X}_n) \mathbf{Z}_n + \mathbf{Y}_n(\mathbf{X}_n) - B_n(\mathbf{X}_n)]^{+}$$ where $\mathbf{X}_n$ is a Markov chain, $R_n$ is a random or deterministic matrix (often diagonal or composed of contraction mappings), and noise/arrival/service processes are state-dependent.

Solving for transient or stationary transforms in this context involves:

• Conditioning on the state
• Applying Wiener–Hopf factorization or analytic continuation to vector/matrix transforms
• Using recursive substitution and truncation for infinite-dimensional systems (when needed) (Dimitriou, 2 Jul 2025).

Uniqueness and existence of solutions hinge on contraction properties (all $a_i$ strictly inside the unit disk) and regularity of the underlying Markov chain.

5. Special Regimes and Explicit Examples

Examples analyzed include:

• Negative multiplicative factors with rational LSTs for inputs (Boxma et al., 2020)
• Uniformly distributed $V_n$ and exponential $A_n$
• Binary $V_n$ values (±1) and state-dependent increments (Dimitriou, 28 Aug 2025)
• Laplace-distributed jumps, where explicit recursive formulas for density and exit time can be computed numerically for all parameter regimes (Lucrezia et al., 2023)

In Markov-modulated settings, explicit computation of means, variances, and tail asymptotics follows differentiation and perturbation of the vector LST fixed-point equation, often yielding exponential decay in the tails.

6. Fluctuation-Theoretic and Lindley-Type Decompositions

Recent research exploits fluctuation-theoretic decompositions and probabilistic identities involving sums and maxima of Lévy processes observed via Poisson inspection: $$\left( \bar{Y}(T_\beta), G(T_\beta) \right) \sim \left( \bar{Y}(T_{\beta+\omega}), G(T_{\beta+\omega}) \right) + \left( S_{N_{\beta,\omega}}, G_{N_{\beta,\omega}} \right)$$ where $(\bar{Y}(T_\beta), G(T_\beta))$ is the running maximum and its epoch for a Lévy process up to exponential time $T_\beta$, and $S_{N_{\beta,\omega}}, G_{N_{\beta,\omega}}$ pertain to discrete Poisson inspection regimes (Boxma et al., 2022).

Elementary proofs connect these decompositions directly to recursive Lindley-type structures, now generalized to capture the effect of discrete inspection frequency, leading to correction terms and new classes of distributional identities.

7. Arithmetic and Dynamical Extensions

In ergodic theory and arithmetic dynamics, “multiplicative recurrence” is now characterized by necessary and sufficient conditions on arithmetic patterns: $\liminf_{n\to\infty} |f(an+b) - f(cn+d)| = 0$ for all completely multiplicative $f:\mathbb{N} \to \mathbb{S}^1$, with recurrence only possible when $a=c$ and $b=d$ or $a \mid bd$ (Charamaras et al., 4 Dec 2024). This property governs recurrence sets under $(\mathbb{N}, \times)$-actions, linking Lindley-type phenomena with rigid arithmetic-combinatorial constraints.

A plausible implication is that multiplicative Lindley-type recursions studied in ergodic and dynamical systems (where transitions update via multiplicative group actions) inherit recurrence and rigidity phenomena dictated by the underlying number-theoretic structure.

8. Applications and Impact

The multiplicative Lindley recursion framework is central to:

• Queueing models with batch or modulated arrivals, retrials, and reneging
• Reflected AR(1) and higher-order autoregressive time series in economics, finance, and population dynamics
• Fluctuation analysis for Lévy and Markov processes under inspection or partial observation
• Branching process theory, especially in evincing power–law tail behavior by maximal inequalities (Jelenkovic et al., 2014)
• Arithmetic recurrence in dynamical systems and multiplicative ergodic theory

Explicit formulas for transforms, densities, and exit probabilities enable direct numerical evaluation, as in the open-source implementations (Lucrezia et al., 2023), supporting analytical and applied studies in diverse domains.

Table: Selected Multiplicative Lindley Recursion Models

Model description	Multiplicative term $V_n$	Notable features
Scalar AR(1) Lindley process	Random, continuous/discrete	Stability via $\mathbb{E}\log|V_n|$
Markov-modulated scalar or vector recursion	$V_n$ state-dependent, various	Matrix LST functional equations
Recursion with Laplace jumps and reflection	No explicit $V_n$, additive	Recursive density, first-exit time
Poisson-inspected Lévy process decomposition	Induced by discrete obs. rate	Distributional identity, correction
Arithmetic recurrence for $(\mathbb{N},\times)$	Pattern-dictated multiplicative	Rigidity from quadruple condition

9. Concluding Perspective

Multiplicative Lindley’s recursion encapsulates a rich set of probabilistic, analytic, and arithmetic phenomena. Its analysis unifies discrete and continuous stochastic processes, functional equations in transform domains, Markov-modulated dependencies, and dynamical systems with arithmetic constraints. The frameworks developed in (Boxma et al., 2020, Dimitriou, 2 Jul 2025, Dimitriou, 28 Aug 2025, Lucrezia et al., 2023, Boxma et al., 2022, Charamaras et al., 4 Dec 2024) provide rigorous tools for both theoretical investigation and real-world modeling in high-dimensional, modulated, and nonlinear environments, opening avenues for further research in recursive systems with multiplicative structure.