Etemadi's Inequality¶
A maximal inequality bounding excursions of independent partial sums by the worst partial-sum tail probability at a smaller threshold.
Core Idea¶
Let S_k be partial sums of independent real random variables. Etemadi's inequality states that the probability any |S_k| reaches 3α is at most three times the largest probability that a particular |S_k| reaches α.
Its value is structural: a pathwise maximum is controlled without assuming identical distributions or, in the base statement, zero means. Extra moment assumptions and other inequalities can then turn the tail term into convergence results.
Scope of Application¶
- Probability theory. Controls partial-sum maxima.
- Limit theorems. Supports convergence arguments.
- Stochastic processes. Bounds finite-horizon excursions.
- Risk analysis. Relates worst prefix deviation to marginal tails.
Clarity¶
State independence, real-valued increments, n, partial-sum definition, α≥0, absolute values, exact constants, and any additional tail inequality. Do not silently replace max over k with the terminal sum. Inclusion test: Include applications to finite partial sums of independent real variables with the exact threshold and probability constants or a proved equivalent rescaling. Exclusion test: Exclude dependent increments, maxima of individual X_k rather than S_k, vector-valued forms without extension, and Kolmogorov bounds presented as identical. Nearest boundary: Kolmogorov's maximal inequality is a neighbor using zero means and variances; it is not Etemadi's statement. Exit condition: The result exits when independence or the partial-sum maximum is absent. Common misclassifications: It is not a bound for dependent increments without extension. It is not a maximum of the individual X_k. It is not exactly Kolmogorov's inequality. It does not assert equality or optimal constants. Nearest named distinctions: Kolmogorov inequality: Uses variance and centering assumptions in a different bound. Doob inequality: Applies to submartingales. Union bound: Does not exploit this partial-sum structure. Tail bound for S_n: Controls only the terminal sum.
Manages Complexity¶
For Etemadi's inequality, separating Independent variables from Partial sums S_k exposes the first dependency. Relating Threshold α to Constants three then prevents the observed Etemadi's inequality outcome from replacing its defining mechanism.
Abstract Reasoning¶
- For Etemadi's inequality, fix Independent variables and its units or identity.
- Establish how Partial sums S_k functions inside Etemadi's inequality from cited evidence.
- Test Threshold α directly instead of inferring Etemadi's inequality from resemblance.
- Map Running maximum to the defining Etemadi's inequality relation.
- Use Constants three to challenge the closest alternative to Etemadi's inequality.
- Report the Etemadi's inequality boundary, uncertainty, and surviving conclusion.
Knowledge Transfer¶
The event-decomposition strategy transfers to related maximal inequalities when dependence and moment assumptions are reproved. Etemadi's constants do not transfer to martingales or vector sums automatically.
Relationships to Other Abstractions¶
Current abstraction Etemadi's Inequality Domain-specific
Parents (1) — more general patterns this builds on
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Etemadi's Inequality presupposes Boundedness Prime
Etemadi's Inequality presupposes Boundedness because the inequality asserts an explicit upper bound on maximal partial-sum excursions.
Hierarchy path (1) — routes to 1 parentless root
- Etemadi's Inequality → Boundedness
Neighborhood in Abstraction Space¶
Etemadi's Inequality sits in a crowded region of the domain-specific corpus (35th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Combinatorial Optimization & Game Problems (12 abstractions)
Nearest neighbors
- Two-Moment Decision Model — 0.89
- Silverman's game — 0.88
- Misuse of p-values — 0.88
- Concurrent Estimation — 0.88
- Quasimartingale — 0.88
Computed from structural-signature embeddings · 2026-10-08