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PathsMarkov tail bound

Theorem trail

Markov tail bound

The first tail inequality in the concentration ladder.

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Record

Research Record

A compact audit view for the theorem before the full source, proof, diagnostic, and failure-mode sections.

Claim scope
theorem
Evidence level
Exact Lean wrapper
Source precision
reviewed / A
Diagnostics
1 item
Exercises
1 exercise
Failure checks
2 checks
Open Atlas traceOpen evidence recordJump to sourcesJump to failures

Theorem Statement

If the average value of is small, then cannot be large too often. If , then can be at most 1% of the time; otherwise the average would be too high.

Assumptions

  • X is nonnegative
  • X is integrable
  • t > 0

Proof Sketch

Split the expectation over the event where X is at least t. Nonnegativity gives X >= t 1_{X >= t}, so E[X] >= t P(X >= t). Dividing by t gives the bound. No variance, independence, or boundedness enters the argument.

Proof Obligations

Nonnegativity before Markov

assumption

Verify X is nonnegative before dividing by the threshold; the pointwise bound X >= t 1_{X >= t} is the proof step that would fail for signed variables.

Applying Markov to a signed variable can produce a numerically plausible but invalid tail statement.

Threshold is positive

proof step

Check that t > 0 before rearranging E[X] >= t P(X >= t); the inequality direction and division step depend on the threshold sign.

Zero or negative thresholds turn the displayed bound into either a division error or a vacuous claim.

Section-level source support

source check

Use the Wainwright Section 2.1.1 concentration reference as canonical support for the Markov tail-bound pattern before promoting variants.

Without a reviewed locator, nearby concentration statements can be mistaken for the exact theorem being used.

Mean-only tail control is a loose baseline

boundary

Use Markov as a mean-only certificate: it ignores variance, boundedness, independence, and distribution shape, so sharper assumptions should route to Chebyshev, Hoeffding, or Bernstein-style tools.

Treating Markov as a calibrated concentration estimate can make a weak polynomial tail bound look stronger than the evidence permits.

Signed variables need a nonnegative transform

boundary

For signed quantities, apply Markov only after replacing the target by a nonnegative transform such as an absolute value, square, or exponential.

Applying the theorem directly to a signed centered variable is a common route to a false concentration statement.

Source Locators

High-Dimensional Statistics (2019)

proof

Chapter 2 — Basic tail and concentration bounds

wainwright-2019:ch2

kind
chapter
review
reviewed / A
extract
manual

Chapter-level canonical support for Markov-style tail bounds.

High-Dimensional Statistics (2019)

proof

Section 2.1.1 — From Markov to Chernoff

wainwright-2019:sec2.1

kind
section
review
reviewed / A
extract
manual
number
2.1.1

Anchor: From Markov to Chernoff

Reviewed section-level support for the Markov-to-Chernoff concentration pattern used by the Markov tail-bound trail.

Curated Trace

  1. 1
    Tail eventobject

    A threshold on a random variable creates the event whose probability must be bounded.

    Checkpoint

    Check that {X >= t} is measurable before applying probability rules.

  2. 2
    Expectation lower boundcalculation

    Expectation turns pointwise nonnegativity into an average inequality.

    Checkpoint

    Verify X >= t 1_{X >= t}; this is where nonnegativity matters.

  3. 3
    Markov inequalitytheorem

    The lower bound rearranges directly into a probability tail bound.

    Checkpoint

    The result uses only E[X], so the tail decay is only 1/t.

  4. 4
    Variance upgradenext theorem

    Applying Markov to a squared centered deviation produces Chebyshev.

    Checkpoint

    Ask what extra information variance buys before moving to Hoeffding.

Failure Modes

Mean-only tail control is a loose baseline

Use Markov as a mean-only certificate: it ignores variance, boundedness, independence, and distribution shape, so sharper assumptions should route to Chebyshev, Hoeffding, or Bernstein-style tools.

Risk: Treating Markov as a calibrated concentration estimate can make a weak polynomial tail bound look stronger than the evidence permits.

Check: Use Markov as a mean-only certificate: it ignores variance, boundedness, independence, and distribution shape, so sharper assumptions should route to Chebyshev, Hoeffding, or Bernstein-style tools.

Signed variables need a nonnegative transform

For signed quantities, apply Markov only after replacing the target by a nonnegative transform such as an absolute value, square, or exponential.

Risk: Applying the theorem directly to a signed centered variable is a common route to a false concentration statement.

Check: For signed quantities, apply Markov only after replacing the target by a nonnegative transform such as an absolute value, square, or exponential.