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CLT limitations

The usual IID CLT needs independent identically distributed draws with finite positive variance, and gives an asymptotic result.

2 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Understand the idea
  3. A closer look
  4. Heavy tails and infinite variance are different claims
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections
Copying a single random draw into several entries does not create independent repetitions. One source feeds three identical copies; this illustrates a failure of the usual IID CLT assumptions, not a guarantee from sample size.Copying a single random draw into several entries does not create independent repetitions. One source feeds three identical copies; this illustrates a failure of the usual IID CLT assumptions, not a guarantee from sample size.
Figure 1Copying a single random draw into several entries does not create independent repetitions. One source feeds three identical copies; this illustrates a failure of the usual IID CLT assumptions, not a guarantee from sample size.
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01Understand the idea#

The usual IID CLT needs independence, identical distributions and finite positive variance. A large count proves none of these.

Copying one nonconstant random value many times creates many entries but no independent repetitions. Alternating between different generating distributions also leaves the stated IID setting.

Failure of this particular theorem’s assumptions does not rule out every other limit theorem.

Check your reasoning

One nonconstant random draw is copied 20 times. Which IID condition fails?

  1. AIdentical marginals
  2. BFinite variance
  3. CIndependence
Show answer and explanation
Independence

The copies share one draw, so they are dependent.

02A closer look#

Even when the assumptions hold, approximation accuracy depends on the distribution and the sample size. Rare extreme values can slow convergence.

There is no universal n=30 rule. A supplied probability comparison can check an accuracy target at that cutoff; it does not certify the entire CDF.

Check your reasoning

True CDF at a cutoff is 0.720.72; normal CDF is 0.70.7. Is error at most 0.030.03 there?

  1. AYes
  2. BNo
  3. CCount alone decides.
Show answer and explanation
Yes

∣0.72−0.7∣=0.02≤0.03|0.72-0.7|=0.02\le0.03 at this cutoff.

03Heavy tails and infinite variance are different claims#

Heavy tails and infinite variance are different claims. If finite positive variance is given, the asymptotic CLT applies, though finite-sample accuracy can be poor.

If the population variance is explicitly infinite, this usual finite-variance CLT is not licensed. More draws do not change that population property.

Check your reasoning

IID, finite positive variance, heavy right tail. Report: exact normality at n=30. Repair it.

  1. AExact normality now.
  2. BAsymptotic CLT.
  3. CInfinite variance.
Show answer and explanation
Asymptotic CLT.

Finite-n accuracy remains unchecked.

Key takeaway

Check sampling and variance assumptions, then assess finite-sample accuracy separately.

  • Diagnose a normal approximation undermined by the sampling assumptions or heavy tails.

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “CLT limitations.” Math behind ML. /learn/p-clt-conditions