Lecture 38 — Lindeberg–Feller CLT via Characteristic Functions and Truncation
This lecture gives:
The full Taylor expansion bounds for characteristic functions,
A full CF-based proof of the Lindeberg–Feller CLT,
A deep truncation example where the variance is infinite but CLT holds after normalization by 𝑛 log 𝑛 nlogn
,
And finally Paul Lévy’s criterion for the domain of attraction of the normal law.
This lecture returns to the characteristic function method, proving the Lindeberg–Feller CLT directly from Taylor expansion.
We then analyze a heavy–tailed example where variance is infinite, showing how truncation recovers a CLT with a nonstandard normalization.
1. Key Taylor Bounds for $e^{ix}$
(Page 1 begins with recalling the fundamental bounds.)
For all real $x$:
\[\vert e^{ix} - 1\vert \le \vert x\vert \wedge 2.\]This follows from:
\[\vert \int_0^x e^{iu}\,du\vert = \vert \frac{e^{ix}-1}{i}\vert \le \vert x\vert .\]Second-order bound:
\[\vert e^{ix} - (1 + ix)\vert \le \frac{x^2}{2} \wedge 2\vert x\vert .\]Third-order bound (used for CLT):
\[\boxed{ \vert e^{ix} - \left(1 + ix - \frac{x^2}{2}\right)\vert \,\le\, \frac{\vert x\vert ^3}{6} \wedge x^2. } \tag{1}\]This 3rd-order control is the backbone of the Lindeberg–Feller CF proof.
2. Apply Taylor to Characteristic Functions
Replace $x$ with $tX$, take expectations:
\[\varphi_X(t) = E[e^{itX}].\]Using (1):
\[\boxed{ \vert \varphi_X(t) - \left(1 + it\,E[X] - \frac{t^2E[X^2]}{2}\right) \vert \le E\!\left[ \frac{\vert tX\vert ^3}{6} \wedge t^2 X^2 \right]. } \tag{2}\]If $E[X]=0$, this simplifies to:
\[\vert \varphi_X(t) - \left(1 - \frac{t^2E[X^2]}{2}\right)\vert \le E\!\left[ \frac{\vert t\vert ^3\vert X\vert ^3}{6} \wedge t^2 X^2 \right]. \tag{3}\](Page 1 highlights this with the star $(*)$.)
3. Lindeberg–Feller CLT via Characteristic Functions
We have a triangular array:
\[\{X_{n,m}\}_{1\le m\le n}, \qquad E[X_{n,m}] = 0, \qquad \sum_{m=1}^n E[X_{n,m}^2] = 1.\]Let:
\[S_n = \sum_{m=1}^n X_{n,m}, \qquad \varphi_n(t) = E[e^{itS_n}].\]Define:
\[Z_{n,m}(t)=\varphi_{X_{n,m}}(t), \qquad \sigma_{n,m}^2 = E[X_{n,m}^2].\]We approximate each $Z_{n,m}(t)$ by a “Gaussian-like” term:
\[w_{n,m}(t) = 1 - \frac{t^2\sigma_{n,m}^2}{2}. \tag{4}\]Since $\sigma_{n,m}^2\to0$ uniformly under Lindeberg (recall from Lecture 35:
$\max_m\sigma_{n,m}^2\to0$), we may treat (4) as a good 2nd-order CF approximation.
3.1 Bounding the error term
(Page 1 bottom → page 2 top.)
Using (3):
\[\vert Z_{n,m}(t) - w_{n,m}(t)\vert \le E\!\left[ \frac{\vert t\vert ^3\vert X_{n,m}\vert ^3}{6} \wedge t^2 X_{n,m}^2 \right].\]Split on $\vert X_{n,m}\vert \le \varepsilon$:
-
If $\vert X_{n,m}\vert \le \varepsilon$, then
$\vert X_{n,m}\vert ^3 \le \varepsilon X_{n,m}^2$. -
If $\vert X_{n,m}\vert >\varepsilon$, use the Lindeberg term.
Thus:
\[\vert Z_{n,m}-w_{n,m}\vert \le \frac{\varepsilon \vert t\vert ^3}{6} E[X_{n,m}^2] + t^2 E[X_{n,m}^2;\,\vert X_{n,m}\vert >\varepsilon]. \tag{5}\]Sum over $m=1,\dots,n$:
\[\sum_{m=1}^n \vert Z_{n,m}-w_{n,m}\vert \le \frac{\varepsilon \vert t\vert ^3}{6} + t^2 L_n(\varepsilon), \tag{6}\]where
\[L_n(\varepsilon) = \sum_{m=1}^n E[X_{n,m}^2;\ \vert X_{n,m}\vert >\varepsilon].\]Under Lindeberg: $L_n(\varepsilon)\to0$.
Thus the RHS of (6) → $\varepsilon \vert t\vert ^3/6$.
Then send $\varepsilon\to0$.
(Page 2: the full computation matches this.)
3.2 From products to sums
Characteristic function of sum:
\[\varphi_n(t) = \prod_{m=1}^n Z_{n,m}(t).\]Since $\vert Z_{n,m}\vert \le1$ and $\vert w_{n,m}\vert \le1$, the inequality (page 2):
\[\vert \prod_{m=1}^n Z_{n,m}-\prod_{m=1}^n w_{n,m}\vert \le \sum_{m=1}^n \vert Z_{n,m}-w_{n,m}\vert \tag{7}\]holds.
But
\[\prod_{m=1}^n \left(1 - \frac{t^2\sigma_{n,m}^2}{2}\right) \to \exp\left(-\frac{t^2}{2}\right), \tag{8}\]because $\sum \sigma_{n,m}^2 =1$ and $\max_m\sigma_{n,m}^2\to0$.
(Page 2 includes the standard lemma: if $\sum a_{n,m}\to a$, $\max \vert a_{n,m}\vert \to0$, then $\prod (1+a_{n,m})\to e^a$.)
Thus:
\[\varphi_n(t) \to e^{-t^2/2}.\]By Continuity Theorem,
\[S_n \Rightarrow N(0,1).\]This proves the Lindeberg–Feller CLT via characteristic functions.
4. Heavy-Tailed Example: Truncation Necessary
(Pages 2–3.)
Let $X_1,X_2,\dots$ be iid, symmetric ($X\overset d= -X$), with tail:
\[P(\vert X\vert \ge x)=x^{-2}, \qquad x\ge1,\]and $P(\vert X\vert <1)=0$.
Then:
- $E[X]=0$ (symmetry),
- $E[X^2]=\infty$ (see page 2 computation),
- No classical CLT can hold under $\sqrt{n}$ scaling.
We apply truncation.
Let
\[C_n = \sqrt{n\log\log n}, \qquad Y_{n,m} = X_m\,\mathbf{1}_{\vert X_m\vert \le C_n}.\]Define:
\[S_n = \sum_{m=1}^n X_m, \qquad T_n = \sum_{m=1}^n Y_{n,m}.\]4.1 Approximation quality
(Page 3 top.)
\[P(S_n \ne T_n) \le n\,P(\vert X\vert >C_n) = \frac{n}{C_n^2} = \frac{1}{\log\log n} \to 0.\]So $S_n$ and $T_n$ differ with vanishing probability.
4.2 Compute variance of $T_n$
Since density behaves like $2y^{-3}$ for large $y$, the truncated second moment is:
\[E[Y_{n,m}^2] = \int_{1}^{C_n} y^2 \cdot \frac{2}{y^3}\,dy = 2\int_1^{C_n} y^{-1}\,dy = 2\log C_n \sim \log n.\]Hence:
\[\mathrm{Var}(T_n) = n\,\mathrm{Var}(Y_{n,m}) \sim n\log n.\](Page 3 matches this exactly: “Var($T_n$) = n\log(n)”.)
4.3 Apply Lindeberg–Feller to $T_n$
We check Lindeberg for the triangular array $Y_{n,m}/\sqrt{n\log n}$.
The same calculations as above show the Lindeberg condition holds.
Thus:
\[\frac{T_n}{\sqrt{n\log n}} \Rightarrow N(0,1).\]Since $S_n - T_n$ is negligible relative to $\sqrt{n\log n}$, we conclude:
\[\boxed{ \frac{S_n}{\sqrt{n\log n}} \Rightarrow N(0,1). }\]This is a nonstandard CLT, valid even though $E[X^2]=\infty$.
5. Paul Lévy: Domain of Attraction of the Normal Law
(Page 3 bottom.)
A sequence of iid variables $X_k$ is in the domain of attraction of the normal law if there exist sequences $a_n$, $b_n>0$ such that:
\[\frac{S_n - a_n}{b_n} \Rightarrow N(0,1).\]Lévy’s criterion:
\[\boxed{ \frac{y^2\,P(\vert X\vert >y)}{E[X^2;\,\vert X\vert >y]} \;\xrightarrow[y\to\infty]{}\; 0. } \tag{9}\]This is equivalent to being in the Gaussian domain of attraction.
Our example satisfies this with the normalization $b_n=\sqrt{n\log n}$.
Cheat–Sheet Summary — Lecture 38
- Third-order Taylor bound for CFs: \(\vert e^{ix}-(1+ix-x^2/2)\vert \le \vert x\vert ^3/6 \wedge x^2.\)
- CF bound: \(\vert \varphi_X(t)-(1 - t^2E[X^2]/2)\vert \le E[(\vert tX\vert ^3/6)\wedge t^2X^2].\)
- Lindeberg–Feller CLT follows by approximating: \(\prod_{m=1}^n \varphi_{X_{n,m}}(t) \approx \prod_{m=1}^n \left(1 - t^2\sigma_{n,m}^2/2\right) \to e^{-t^2/2}.\)
- Heavy-tail truncation example: \(\frac{S_n}{\sqrt{n\log n}} \Rightarrow N(0,1) \quad\text{even though }E[X^2]=\infty.\)
- Lévy’s necessary and sufficient condition for normal attraction: \(\frac{y^2 P(\vert X\vert >y)}{E[X^2;\vert X\vert >y]}\to 0.\)
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