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Etemadi's Inequality

A maximal inequality bounding excursions of independent partial sums by the worst partial-sum tail probability at a smaller threshold.

Version
v1 · 2026-09-28 · History
Domain-specific #
9308
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Probability Theory, Maximal Inequalities → Mathematics
Aliases
Etemadi maximal inequality

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.

Structural Signature

Sig role-phrases:

  • Independent variables — Supply increments with factorable event structure. It is assumption. Counterfactual: Dependence can invalidate the bound.
  • Partial sums S_k — Accumulate the first k variables. It is process. Counterfactual: A maximum over unrelated variables is not this inequality.
  • Threshold α — Sets the comparison scale. It is parameter. Counterfactual: Negative α is outside the stated form.
  • Running maximum — Captures whether any prefix has a large absolute deviation. It is target event. Counterfactual: Only checking S_n misses earlier excursions.
  • Largest marginal tail — Provides the right-hand comparison across k. It is bound input. Counterfactual: Replacing maximum by one arbitrary k can underbound risk.
  • Constants three — Relate outer threshold and probability multiplier. It is quantitative form. Counterfactual: Changing constants requires another theorem or proof.

What It Is Not

  • 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.
  • Closest near-miss. Kolmogorov's maximal inequality is a neighbor using zero means and variances; it is not Etemadi's statement.

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.

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

  1. For Etemadi's inequality, fix Independent variables and its units or identity.
  2. Establish how Partial sums S_k functions inside Etemadi's inequality from cited evidence.
  3. Test Threshold α directly instead of inferring Etemadi's inequality from resemblance.
  4. Map Running maximum to the defining Etemadi's inequality relation.
  5. Use Constants three to challenge the closest alternative to Etemadi's inequality.
  6. 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.

Examples

Applied / In Practice

For independent X_1,…,X_n, compute tails of each S_k at α; three times their maximum controls the chance any |S_k| reaches 3α.

Mapped back: increments → independent; target → max partial sum; outer → 3α; bound → 3 max tails.

Applied / In Practice

Strongly dependent increments are substituted without proof, so marginal partial-sum tails no longer justify Etemadi's maximal bound.

Mapped back: assumption → dependence; missing → independence; status → not licensed.

Structural Tensions

T1 — Maximum Path Event versus Marginal Tail Information. The theorem controls an entire prefix path using only the worst single-time tail.

Diagnostic: Are the events genuinely built from independent increments?

T2 — Simple Constants versus Sharpness. The factors three offer broad assumptions rather than an optimized bound for every distribution.

Diagnostic: Is a sharper theorem needed for the decision?

Structural–Framed Character

Within Etemadi's inequality, the relation among Independent variables, Partial sums S_k, and Threshold α forms the structural core; Constants three supplies the decisive condition for Etemadi's inequality.

Structural Core vs. Domain Accent

The Etemadi's inequality identity is distinguished by how Running maximum constrains Constants three; their pairing anchors vocabulary to evidence specific to Etemadi's inequality.

This entry presupposes Boundedness.

  • Approved root. This independent-sum maximal bound has no frozen parent.

  • Related — Kolmogorov inequality, Doob inequality, partial sum, and law of large numbers. They are neighboring bounds, objects, or applications.

Relationships to Other Abstractions

Local relationship map for Etemadi's InequalityParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Etemadi's InequalityDOMAINPrime abstraction: Boundedness — presupposesBoundednessPRIME

Current abstraction Etemadi's Inequality Domain-specific

Parents (1) — more general patterns this builds on

  • 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

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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Kolmogorov inequality. Tell: Uses variance and centering assumptions in a different bound.
  • Doob inequality. Tell: Applies to submartingales.
  • Union bound. Tell: Does not exploit this partial-sum structure.
  • Tail bound for S_n. Tell: Controls only the terminal sum.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Etemadi%27s_inequality (revision 1281696550).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.