The study of quasi-stationary distributions (QSDs) originated in the mid-20th century to describe systems that persist in transient states before eventual absorption. Although developed within branching, diffusion, and Markov processes, the framework naturally extends to promotional dynamics where temporary stability precedes termination. ^{2, 6, 8– 10} The historical progression of quasi-stationary research across foundational decades is summarized in Figure 2.

Yaglom (1947) analyzed subcritical branching processes and showed that conditional distributions converge to stable forms given non-extinction. ² Bartlett (1955) and Kingman (1963) expanded spectral interpretations linking decay rates to generator eigenvalues. ^{6, 8}

Darroch and Seneta ( 1965 , 1967 ) formalized QSDs for continuous-time Markov chains, proving existence and uniqueness and establishing the canonical relationship mQ = − αm . ³ Vere-Jones (1969) developed limit theorems and invariance properties, ¹¹ while Pakes (1973) connected QSDs to random-walk and catastrophe models. ⁷

From 1991 to 1994, Van Doorn and collaborators introduced numerical methods for quasi-birth-death (QBD) processes, enabling large-scale computation of ( m , α ) and bridging stochastic theory with real-world analytics. ^{12, 13}

A chronological summary of these foundational contributions is provided in Table 2.

QSD theory is well established in probability and applied stochastic processes, but there is no prior work applying quasi-stationary analysis to trade promotion effectiveness, conditional sales behavior, or marketing lift persistence. Existing CPG promotion models focus on lift magnitude, elasticity, or carryover effects, whereas QSDs characterize the conditional composition and duration of promotional states before termination. This gap is where our contribution comes in; introducing ( m , α ) as interpretable stochastic indicators of promotional persistence and mechanic dominance.

Promotional dynamics in CPG markets can be rigorously modeled as a continuous-time Markov process, where each week corresponds to a transition among distinct promotional states. Let the finite state space be S = { TPR , Feature , Display , Feature + Display , End } where “End” denotes the absorbing state representing promotion termination, and all other states form the transient subset S ′ = S \ { End } .

The process evolves according to a transition-rate matrix Q = ( q ij ) , where q ij ≥ 0 , i ≠ j , q ii = − ∑ j ≠ i q ij

The corresponding transition-probability matrix at time t is P ( t ) = e Qt , where each entry p ij ( t ) gives the probability of moving from state i to j after t weeks. ⁸ Because every promotion eventually ends, the absorbing state is reached with probability one: Pr [ X ( t ) = End ] → 1 as t → ∞ .

A distribution m = ( m i ) , i ∈ S ′ is quasi-stationary if the conditional distribution over the active states remains constant given that the promotion has not yet ended: mP ( t ) = c ( t ) m , t > 0 where c ( t ) is the probability that the promotion remains active at time t .

Differentiating (5) at t = 0 yields the key eigenvalue condition: mQ = − αm , where α > 0 is the decay rate representing the exponential hazard of promotional termination. ³

•

The vector m represents the long-run conditional mix of promotion mechanics (TPR, Feature, Display, Feature + Display) given that the promotion is still active.

Together, these components form a rigorous stochastic structure that captures the balance between temporary stability and inevitable decline within a promotional campaign. ^{2, 6}

The transition structure among these states is illustrated in Figure 3. A detailed summary of notation and state definitions is provided in Table 3.

Promotional programs typically cycle through multiple mechanics before termination. A product may move from a TPR to a feature advertisement, subsequently receive a display, or combine both tactics in a feature + display campaign before eventually reverting to base pricing or ending the promotion. ¹ This evolving structure is captured through a transition network of promotional states with distinct lift patterns.

Let the transient state set be S ′ = { 1 , 2 , 3 , 4 } corresponding to TPR, Feature, Display, and Feature + Display. For active states i , j ∈ S ′ , the one-step transition probability is q ij = Pr [ X ( t + 1 ) = j | X ( t ) = i ] , i ≠ j .

The self-transition q ii represents the probability that a promotion remains in the same configuration for an additional week. Empirically estimating these probabilities across weeks yields the transition sub-matrix Q ( 0 ) that governs pretermination behavior. ³

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TPR → Feature/Display: escalation from price-only tactics toward awareness-building mechanisms.

These transitions shape both the persistence probability and the conditional composition vector m in the quasi-stationary regime.

The empirical transition matrix is estimated as the proportion of weeks in which a promotion moved from state i to j in the simulated 104-week dataset: q ^ ij = Number of transitions from i to j Total transitions from i .

The resulting matrix Q ( 0 ) is row-sub-stochastic: each row sums to less than one, with the remainder representing transition to the absorbing “End” state. ⁶ These probabilities feed directly into the eigenvalue decomposition used to derive ( m , α ), as discussed in Section.

A visual representation of week-to-week transitions among promotional mechanics is shown in Figure 4. An illustrative set of weekly transition probabilities is provided in Table 4.

The quasi-stationary framework quantifies promotional persistence and the conditional composition of active promotion types. Operationally, it estimates how long a promotion remains effective before termination and the distribution of promotion mechanics during its active life. ¹

Let Q ( 0 ) denote the sub-generator among transient states (excluding the absorbing End state). If the transition matrix among active states is written as P ( 0 ) ( t ) = e Q ( 0 ) t , then the taboo survival function for a promotion starting in state i is: S i ( t ) = ∑ j ∈ S ′ p ij ( 0 ) ( t ) , S i ( t ) ∼ C i e − αt .

The exponential decay implies that the probability of a promotion still being active declines at rate α , the dominant eigenvalue of Q ( 0 ) . ³ The corresponding left eigenvector m defines the quasi-stationary distribution: m Q ( 0 ) = − αm , ∑ i m i = 1 .

The expected duration of promotional effectiveness follows: E [ T ] = 1 α

As shown earlier in Figure 3, this structure captures how promotions transition among mechanics before eventual termination.

Let λ 1 , λ 2 , … , λ n denote the eigenvalues of Q ( 0 ) , ordered such that: R ( λ 1 ) > R ( λ 2 ) ≥ ⋯ ≥ R ( λ n ) .

The principal eigenvalue λ 1 = − α determines the slowest decaying component of the process and governs long-term survival behavior. ^{3, 5, 6} The decay rate may be characterized through the functional determinant: | I − G α ( p ) | = 0 which identifies the dominant root associated with promotional decay. ⁷

We estimate ( m , α ) using simulated multi-category promotional data. Each category reflects distinct dynamics in promotional persistence, decay rate, and conditional state composition. ¹

Weekly UPC-retailer promotional histories are used to construct the empirical transition sub-generator Q ( 0 ) . Eigenvalue decomposition produces the decay rate α and the quasi-stationary composition vector m . The probability that a promotion remains active after t weeks is given by the survival function in (14): S ( t ) = e − αt .

The associated expected promotional lifespan is E [ T ] = 1 / α . ³

Feature + Display typically exhibits the lowest decay rate α (longer promotional persistence), TPR is moderate, and Display-only often yields higher α values (faster decay). Categories such as beverages and dairy show long-lived promotional cycles, while certain household categories exhibit more rapid decline. ¹²

The decay rate α supports decisions on optimal promotion duration. Campaigns with small α benefit from extended runs, whereas campaigns with large α are best rotated or terminated quickly. The QSD vector m provides tactical allocation weights for funding promotion types that sustain conditional lift.

The decay profile and corresponding survival function are illustrated in Figure 6.

Table 6 summarizes estimated parameters across representative categories.

We construct a 104-week simulated UPC-retailer dataset with weekly transitions among TPR, Feature, Display, Feature + Display, and the absorbing End state. This controlled environment enables validation of the quasi-stationary estimation of ( m , α ) .

Each weekly observation includes promotional state and sales. Transitions follow a predefined matrix ensuring eventual absorption, emulating real-world cycles in which lift diminishes after repeated exposure. ¹ A subset of the simulated 104-week promotional dataset is shown in Table 7.

Let n ij denote transitions from state i to j . The empirical transition probabilities are estimated as: q ^ ij = n ij ∑ j ∈ S ′ n ij .

The quasi-stationary parameters follow from the eigen-decomposition: m Q ( 0 ) = − αm , ∑ i m i = 1 .

The survival curve follows: S ( t ) = Pr [ T > t ] = e − αt .

The corresponding empirical survival curve and its fitted exponential decay are shown in Figure 7.

The transition intensities across active promotion mechanics are visualized in the heatmap shown in Figure 8.

Empirical-theoretical alignment validates QSD applicability for promotional persistence. ¹² Longevity depends not only on duration but also on tactical transitions, complementing machine-learning uplift and BSTS models used in planning dashboards.

Translating ( m , α ) into operational decisions enables optimization of promotion duration, mechanic rotation, and return on investment (ROI). ¹ A visual summary of these decision zones, based on the decay rate α , is shown in Figure 9.

A lower decay rate indicates longer promotional persistence. •

α between 0.10 and 0.14: long-lived promotions.

Even small increases in α significantly reduce the expected lifespan E [ T ] = 1 / α .

The quasi-stationary composition m indicates which tactic dominates conditional effectiveness: •

High m F + Display : display-sensitive categories-invest in co-located displays.

Decision zones include Sustain (low α ), Rotate (moderate), and Terminate (high). ¹² These rules can be integrated into enterprise dashboards for stop/continue signals (based on α ), expected duration ( 1 / α ), and mechanic mix ( m i ).

A detailed summary of decision rules by decay rate, expected lifespan, and ROI implications is provided in Table 10.

Quasi-stationary metrics integrate naturally into enterprise analytics platforms (e.g., Circana’s Demand Forecaster or Promo Effectiveness Dashboard). Key indicators include: •

α : hazard of promotional fatigue (stop-continue signal)

Aligning these indicators with profitability and media effectiveness enables stochastic decision optimization: a discipline that quantifies persistence, lift decay, and mechanic efficiency in a unified framework. ^{3, 14}