LLN Failure Under Infinite Mean

Statement¶

Let $X_1,X_2,\dots$ be independent copies of a nonnegative random variable $X$, and let

If $\mathbb E[X]=\infty$, then

For a Pareto Type I variable with lower cutoff $x_m>0$ and tail exponent $\alpha$, this applies exactly when $\alpha\le1$. In that regime the sample mean is not estimating a hidden finite number; the ordinary law of large numbers normalization has failed.

Here $\mathbb E$ is expectation and a.s. means almost surely. The Pareto parameters $x_m$ and $\alpha$ are the lower cutoff and positive tail exponent. See Shared notation.

Intuition¶

The finite-mean law of large numbers says that many small and moderate observations eventually average out the noise. With a nonnegative infinite mean, every finite truncation has an ordinary average, but those truncation levels can be pushed higher without bound. The untruncated average must eventually dominate each one.

For Pareto tails with $\alpha\le1$, rare observations are large enough that no finite long-run mean exists. A running average may drift downward between records, but the theorem says there is no stable finite level waiting in the limit.

Proof sketch¶

We prove the idea by truncating the variables, applying the strong law to each bounded truncation, and then letting the cutoff rise. The Pareto threshold condition is not reproved here; it is supplied by Pareto Moment Existence.

For a cutoff $c>0$, define the truncated variable

Each $X_i^{(c)}$ is bounded, so the strong law of large numbers gives

Since $X_i\ge X_i^{(c)}$,

almost surely for each fixed $c$. Apply this on the countable sequence $c=1,2,3,\dots$. By monotone convergence,

Therefore the liminf of $S_n/n$ is larger than every finite number, almost surely, which proves $S_n/n\to\infty$ almost surely.

For a Pareto Type I variable, Pareto Moment Existence proves that $\mathbb E[X]$ is infinite exactly when $\alpha\le1$.

Sample-path behavior¶

The running average obeys the exact update identity

It falls whenever the next observation is below the current average and rises whenever it is above. Divergence to infinity does not imply a monotone sample path. Long downward stretches between records are compatible with the theorem. The max-to-sum ratio measures a different quantity: the largest observation’s share of the realized total. In particular, infinite mean alone does not imply that this share tends to one.

Capped means for exact Pareto tails¶

The truncation proof can be inspected directly. For Pareto Type I with $x_m=1$ and $c\ge1$,

The following values are evaluations of this identity, rounded to three decimals. They are capped population means, not sample averages or $\mathbb E[X\mathbf 1_{\{X\le c\}}]$.

For a general regularly varying tail at exponent $\alpha=1$, the slowly varying factor can make the mean finite. The theorem requires infinite mean; the exact Pareto cutoff $\alpha\le1$ is not a universal boundary rule at equality. See Karamata’s theorem for the tail-integral criterion.

Counterexamples and caveats¶

• We state a one-sided nonnegative result. If a distribution has large positive and negative tails, failure of the ordinary LLN can mean non-convergence rather than divergence to $+\infty$.

• The theorem is asymptotic. A finite sample from an infinite-mean law can still show long quiet stretches, especially before the next record-sized observation arrives.

• A finite mean does not guarantee a comfortable sample size. When $1<\alpha<2$, the mean exists but the variance is infinite, so convergence of the sample average can still be painfully slow.

• Dependence, truncation, censoring, and changing thresholds can alter what a real data set shows. This result is the iid baseline.

References¶

• Feller, An Introduction to Probability Theory and Its Applications, Vol. II 1971, 2nd ed., Wiley.

• Resnick, Heavy-Tail Phenomena 2007.

Backlinks¶

• Depends on: Pareto Distribution and the Shared notation.

• Used by: Pre-Asymptotic LLN Behavior and Max-to-Sum Ratio.