Theorem 11 (stretched-exponential survival bound)

Apparatus §9 · rate-of-foreclosure under intermediate tail conditions

The rate-of-foreclosure refinement. Lemma 1 gives polynomial decay under bounded coefficient of variation and exponential decay under finite moment-generating function. Theorem 11 fills the gap: under Weibull-shape sub-exponential tails (Class $S_{β}$ for $β \in (0, 1]$ ), survival decays as $exp (- J_{β} (θ_{ref}^{β}))$ — a stretched exponential.

Lemma 1 has two regimes: polynomial decay under bounded $c_{v}$ (unconditional, but loose) and exponential decay under finite MGF (sharp, but requires heavier tail condition). The intermediate regime — sub-exponential or stretched-exponential tails — is where most empirical engagement-interval distributions sit (Weibull, log-normal, gamma-with-shape less than one). §9 supplies the missing bound.

The stretched-exponential class. An inter-arrival distribution belongs to Class $S_{β}$ for $β \in (0, 1]$ if $E [exp (s τ^{β})]$ is finite on some interval $[0, s_{0})$ . Class $S_{1}$ recovers the standard MGF condition. Class $S_{β}$ for $β < 1$ captures Weibull-shape tails: $E [exp (s τ)]$ may be infinite, but $E [exp (s τ^{β})]$ is finite.

The empirical-tail question. Inter-arrival times for engagement events on real platforms have been measured across multiple studies, with consistent findings: log-normal and Weibull-with-shape-less-than-one are the dominant fits, with the shape parameter $β$ typically in the 0.3-to-0.7 range. Stouffer et al. (2006) on email correspondence; Vázquez et al. (2006) on web-browsing intervals; Cheng et al. (2014) on YouTube watch-session lengths — all in the stretched-exponential class. The framework's intermediate-regime bound is therefore the regime most engagement-interval data sits in, which is what gives Theorem 11 its empirical traction.

The interpolation.

Tail class	β	Bound on S(θ_ref; θ*)
Polynomial (bounded c_v only)	—	O((θ_ref/δ_min)^(−2))
Stretched exponential (S_β, β < 1)	β	O(exp(−(θ_ref/σ)^β))
Exponential (Class L = S_1)	1	O(exp(−c · θ_ref/δ_min))

What the theorem authorizes the prose to claim. The foreclosure bound is empirically tight for mature platforms. For $θ_{ref} / δ_{m i n} \sim 1 0^{3}$ and empirically-typical Weibull shape $β \approx 0.5$ , Theorem 11 gives $S ≲ exp (- 31)$ — operationally zero, substantially sharper than the polynomial bound's $1 0^{- 6}$ . The framework's foreclosure claim has substantial buffer under realistic tail conditions; it sits well clear of any knife-edge bound.

The empirical hook. Fit a Weibull tail to the right portion of the empirical CCDF: $lo g (- lo g \hat{S} (τ)) = β lo g τ + const$ . The slope gives the empirical $\hat{β}$ . The framework predicts $\hat{β}$ should increase toward $1$ as platforms mature — a testable prediction about engagement-interval tail-shape evolution on contemporary platforms.

Definition 9.1 (Class S_β — sub-exponential of order β)

An inter-arrival distribution belongs to Class $S_{β}$ for $β \in (0, 1]$ if there exists $s_{0} > 0$ such that

M_{β} (s) := E [exp (s τ^{β})] < \infty for all s \in [0, s_{0}) .

in wordsA distribution joins class $S_{β}$ if its $β$ -stretched moments stay finite — a weaker demand than a finite ordinary moment-generating function (the $β = 1$ case). Smaller $β$ tolerates heavier tails, and the smallest $β$ a distribution satisfies is a label for how heavy its tail is.

The class is monotone in $β$ at the level of inclusion: $S_{1} \subseteq S_{β}$ for $β < 1$ . The smallest $β$ for which a distribution belongs characterizes its tail decay rate.

Theorem 11 (stretched-exponential survival bound)

Suppose the inter-arrival distribution at $θ^{*}$ belongs to Class $S_{β}$ for some $β \in (0, 1]$ . Then

S (θ_{ref}; θ^{*}) \leq exp (- J_{β} (θ_{ref}^{β})),

in wordsSurvival at the threshold decays like a stretched exponential — $exp$ of minus a rate evaluated at the $β$ -power of the threshold. As $β$ sweeps from 0 to 1, this single bound interpolates between the loose polynomial bound of Lemma 1 and the sharp exponential one — and it covers the heavy-tailed regime real engagement-interval data actually occupies.

where $J_{β} (x) := sup_{s \in [0, s_{0})} [s x - lo g M_{β} (s)]$ is the Cramér transform of $τ^{β}$ evaluated at $θ_{ref}^{β}$ .

The bound interpolates between polynomial and exponential regimes as $β$ varies in $(0, 1]$ .

Proof

For $s \in [0, s_{0})$ , Markov on the monotone transformation $τ \mapsto exp (s τ^{β})$ :

Pr (τ > θ_{ref}) = Pr (exp (s τ^{β}) > exp (s θ_{ref}^{β})) \leq M_{β} (s) exp (- s θ_{ref}^{β}) .

in wordsThe proof in one move. A long pause is the same event as its stretched exponential exceeding a large value, and Markov's inequality caps the probability of any positive quantity exceeding a level by its mean divided by that level. This holds for every $s$ .

Infimum over $s$ :

S (θ_{ref}; θ^{*}) \leq exp (- s sup [s θ_{ref}^{β} - lo g M_{β} (s)]) = exp (- J_{β} (θ_{ref}^{β})) .

in wordsSince the bound holds for every $s$ , pick the $s$ that makes it tightest. That optimization is the Cramér transform $J_{β}$ — the large-deviations rate — and it delivers the theorem's bound.

Strict positivity. The derivative at $s = 0$ is $θ_{ref}^{β} - E [τ^{β}]$ , which is positive under structural separation $θ_{ref} / δ_{m i n} ≫ 1$ . Hence $J_{β} (θ_{ref}^{β}) > 0$ .

∎

Cross-references

Refines: Lemma 1 — interpolates between Lemma 1's polynomial bound (b) and exponential bound (c)
Required: §0 axiom (R1) and the structural-separation hypothesis
Empirical hook: §12 (empirical calibration) — fitting $\hat{β}$ from the right tail of the CCDF