Chapter 5 · Subchapter 5

The Cauchy Lesson: Heavy Tails and Infinite Means

Every tool in this chapter has been built to be honest about failure. Now the payoff in reverse: a distribution, entirely legal, whose mean does not exist — and whose averages never concentrate. The proofs are short; the lesson is permanent.

5.5.1The Cauchy law, and why its mean fails

Definition — standard Cauchy law

The law with density f(x) = 1/(π(1+x²)) on ℝ — the shape of a narrow Lorentzian peak, heavy shoulders, and tails falling like 1/x². CDF: F(t) = 1/π · (arctan t + π/2). It arises naturally: the tangent of a uniform angle on (−π/2, π/2) — a beam rotating at uniform angular speed, projected on a wall, is Cauchy. Legitimacy check: f ≥ 0 and ∫ f dx = 1 (the arctan integral — first-year calculus, and by 5.3 no compatibility worry).

Theorem 1 — the Cauchy mean does not exist

For X standard Cauchy, E[X⁺] = E[X⁻] = ∞. By the definition of expectation (5.1 §4), E[X] is the difference of two infinities: undefined. Not large — undefined.

Proof — read the tails off the layer cake

By the layer-cake identity (5.1 Example 4, provable in one MCT line from banding): for Y ≥ 0, E[Y] = ∫₀^∞ P(Y > t) dt. For the positive part: P(X⁺ > t) = P(X > t) = ∫_t^∞ dx/(π(1+x²)) = (1/π)(π/2 − arctan t) ≈ 1/(πt) for large t — the 1/t tail is the whole story. Then

E[X⁺] = ∫₀^∞ P(X > t) dt ≈ ∫^∞ dt/(πt) = ∞

divergent as the harmonic series (5.1 Example 3's series, wearing continuous clothes). Precisely: bound P(X > t) ≥ c/t for t ≥ 1 and integrate. By symmetry E[X⁻] = ∞ too — the law is symmetric, both tails are equally heavy. Verdict per the three-case scheme of 5.1 §5: the mean is in the nonexistent category, the worst of the three. ∎

Why the naive computation lies

The symmetry temptation: "the density is symmetric, so the mean is 0." Symmetry gives E[X] = −E[X] only when E[X] exists — the manipulation is licensed by integrability, and integrability is what failed. The integral ∫_(−∞,∞) x/(π(1+x²)) dx is a classic conditionally-convergent-looking object: its principal value is 0, but as a Lebesgue integral it is +∞ − ∞, and the course's definition (5.1 §4) refuses — deliberately. The refusal is what makes linearity reliable for every integral that does exist. This is the same discipline as 5.1's warning: a computation you cannot justify is a number you do not own. "Mean 0 by symmetry" for Cauchy is the canonical false sentence of the subject.

5.5.2Averages that refuse: X̄ stays Cauchy

Theorem 2 — the sample mean of Cauchys is Cauchy (statement)

If X₁, X₂, … are independent with the standard Cauchy law, then the sample average X̄_n = (X₁+⋯+X_n)/n has the standard Cauchy law — exactly, for every n. Averaging does not concentrate, does not narrow, does not improve. (Independence is defined properly in 6.1; the theorem's proof via characteristic functions comes in 7.3 — e^−|t| raised to the n-th power, divided by n, is again e^−|t|. We state it now because it is the moral's sharpest form.)

Example 1 · Simulation of the failure

Draw 10,000 Cauchy samples and watch X̄_n trace its history: it wanders, occasionally lurches — a single draw of size 10⁶ shifts the running average by ~10³ — and never settles. The standard error-of-the-mean logic from practice presupposes E[X²] < ∞; the Cauchy has E[X²] = ∞ as well: the integrand x² f(x) ~ x²·(1/(πx²)) = 1/(πx) at large |x|, so E[X²] = ∫ x² f(x) dx ≥ ∫₁^∞ dx/(2πx) = ∞ — the same harmonic divergence, one power up. The failure cascades: no second moment, then no first: no second moment, then no first moment. The practitioner's rule "average 100 draws to cut noise by 10×" is a theorem under a hypothesis; the Cauchy kills the hypothesis and the theorem with it. This is the brittleness-flagging habit (1.1) at its most expensive: in production code, an unnoticed 1/x² tail silently converts a "risk estimate" into noise.

5.5.3Tail integrals: the diagnostic

Theorem 3 — moments are tail integrals

For X ≥ 0 and p > 0: E[X^p] = ∫₀^∞ p t^p−1 P(X > t) dt, finite iff the tail integral converges. So the entire moment structure of a law is encoded in the decay rate of P(X > t):

P(X > t) ~ t^−p ⇒ moments of order strictly below p exist; at p and above they fail. (The boundary itself always fails for power tails — the knife-edge of ∫ t⁻¹ dt.)
P(X > t) ~ e^−ct ⇒ all moments exist (light tails).
P(X > t) ~ 1/t ⇒ not even the mean.

Proof — layer cake, iterated

Apply the layer-cake identity to Y = X^p: E[X^p] = ∫₀^∞ P(X^p > s) ds = ∫₀^∞ P(X > s^1/p) ds; substitute s = t^p, ds = p t^p−1 dt — the claimed integral. The equivalence with tail decay is then a comparison test against ∫ t^p−1−p dt-type integrals, exactly the boundary case ∫ t⁻¹ dt that diverges. Note the knife-edge: at the boundary exponent, divergence — there is no "just barely exists" for power tails. ∎

The three regimes of Theorem 3. The vertical axis is survival probability; the question every moment asks is whether the area under the tail curve is finite.

5.5.4The spectrum of tail heaviness

Example 2 · The laws a quant meets, classified by tail

Light tails — Gaussian (e^−t²/2): every moment finite; averages concentrate (7.2 will make this a theorem); extreme observations are effectively impossible — a 6σ event has probability ~10⁻⁹. Power tails — Student-t with ν degrees, Pareto with index α: moments exist strictly below the index, fail at and above. Real equity and crypto returns are empirically in this class (α somewhere between 2 and 5 depending on market and era) — which means: variance exists or not depending on α > 2, kurtosis exists or not depending on α > 4, and estimates of high moments are estimates of things that may not exist. The boundary — α = 1: Cauchy itself. The classification is not academic: it is a checklist that every risk methodology either performs or silently assumes its way past.

Example 3 · Truncated vs genuine: the seduction of finite samples

A Cauchy sample of size 10⁴ has a finite sample mean — every empirical average is a finite number. The failure of Theorem 1 is about the law, not the sample; the sample mean of Cauchys has a distribution (it is Cauchy again, Theorem 2) that does not collapse to a point. So the empirical mean is a random number of full, undiminished spread — and every additional sample "refreshes" it rather than refining it. Contrast with the light-tailed case, where the empirical mean's spread shrinks like 1/√n. The danger in practice: finite samples from heavy-tailed laws look stabilizing over stretches (the running average drifts slowly, then lurches), seducing exactly the inference "it has settled now" that Theorem 2 forbids. Anyone who has watched a crypto strategy's backtest "converge" and then break has met this personally.

5.5.5What this means in practice

The risk checklist, with proofs behind it

Before trusting any mean: estimate the tail index (Hill-style estimators, or simply plot the survival function on log-log axes); if the estimated index is near or below 1, the mean is not a meaningful summary — not "noisy", undefined.
Averaging is not purification: Theorem 2 is a no-go theorem for diversification-under-power-tails at α ≤ 1. Even for 1 < α < 2 the mean exists but the variance does not, and CLT-based confidence intervals (7.3) are unlicensed — the stable-law regime takes over.
Quantiles beat moments for heavy tails: VaR-style summaries (4.2 Example 7) are finite for any tail, since they ask only about fixed levels; expected-shortfall-type summaries reintroduce integrals and inherit the moment risk.
The mathematics is checkable: every claim above reduces to a layer-cake computation you can run yourself — that is the point of the course. No folklore required.

5.5.6Chapter 5, closed

Expectation is built (5.1), limit-safe (5.2), reconciled with calculus (5.3), computable from laws (5.4), and now honestly bounded in its reach (5.5). The number E[X] is neither omnipotent nor optional: it is a tool with a precisely delimited domain, and the course will flag every excursion to the boundary. Chapter 6 turns to the structure that expectation alone cannot see — how several random variables relate: independence as a product-measure concept, Fubini's theorem for computing with it, Borel–Cantelli for the long run of repeated trials, and Bertrand's paradox as the discipline for "random" as an adjective that must always complete the phrase "random with respect to which measure".

The Cauchy law is not a curiosity — it is the price list for forgetting that every averaging theorem has a tail hypothesis.

Where we are, and what is next

One variable at a time, the theory is complete. The next chapter builds the multi-variable world: the product σ-algebra, the theorem that a pair of independent variables is exactly a product measure in disguise, Fubini–Tonelli for exchanging integrals over several variables, and the two results (Borel–Cantelli, Bertrand) that teach what independence really buys and what "at random" really must mean.