Lorden's Inequality¶
Lorden's inequality bounds the mean excess at first crossing of any nonnegative threshold by a positive-part second moment divided by the positive drift of iid increments.
Core Idea¶
For iid increments \(Z_i\) with positive mean \(m\) and finite \(E[(Z^+)^2]\), stop the random walk at its first strict crossing of threshold \(a\geq0\). Lorden's inequality says the mean excess \(R_a\) is at most \(E[(Z^+)^2]/m\), uniformly in \(a\). Nonnegative renewal increments reduce this to \(E[X^2]/E[X]\). The original theorem also admits signed steps.[ref-043b8e053de8][ref-79950f39606f]
Scope of Application¶
In a renewal process, \(R_a\) is the time from an inspection horizon to the next renewal. In sequential analysis, a positively drifting log-likelihood sum may step backward yet still cross an upper evidence boundary; the positive-part form covers that one-sided passage under the theorem's hypotheses. A complete two-boundary test requires additional reasoning.[ref-6a5029794b4e][ref-79950f39606f]
Clarity¶
The threshold \(a\), first-passage count \(T(a)\), and overshoot \(R_a=S_{T(a)}-a\) are different quantities. The inequality bounds an expectation, not every realized excess or its tail.
Manages Complexity¶
One moment ratio controls all thresholds without finding each overshoot distribution. The price is possible looseness for a particular increment law. Positive drift, iid increments and the finite positive-part second moment cannot be discarded.
Abstract Reasoning¶
At crossing, the stopped sum is \(a+R_a\). When Wald's equation applies, \(mE[T(a)]=a+E[R_a]\); Lorden's ceiling then bounds expected crossing count by \(a/m+E[(Z^+)^2]/m^2\).[^ref-79950f39606f]
Knowledge Transfer¶
Renewal lifetimes and signed log-likelihood increments share the first-passage role map, but only the former guarantee nonnegative steps. The live Stopping Time is a strict prerequisite for the overshoot target, not a taxonomic genus of the inequality.
[^ref-043b8e053de8]: Lorden, On Excess over the Boundary, original-paper archive. [^ref-79950f39606f]: Bartroff and Tartakovsky, Lorden theorem exposition, §2.2. [^ref-6a5029794b4e]: Spouge, Overshoot comparison, original research.
Relationships to Other Abstractions¶
Current abstraction Lorden's Inequality Domain-specific
Parents (1) — more general patterns this builds on
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Lorden's Inequality presupposes Stopping time Domain-specific
Lorden's overshoot inequality requires a nonanticipating first-passage stopping time.
Hierarchy path (1) — routes to 1 parentless root
- Lorden's Inequality → Stopping time → Threshold
Neighborhood in Abstraction Space¶
Lorden's Inequality sits in a sparse region of the domain-specific corpus (65th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Foundations of Probability & Inference (29 abstractions)
Nearest neighbors
- Spitzer's Formula — 0.86
- Continuous-Time Random Walk — 0.84
- Yule–Simon Distribution — 0.84
- Arithmetic Progression — 0.84
- Residence Time (Statistics) — 0.84
Computed from structural-signature embeddings · 2026-10-08