Lecture 35 — Lindeberg–Feller CLT and Characteristic Functions
This lecture generalizes the CLT from iid sequences to triangular arrays, introduces the Lindeberg condition, proves the CLT by the one-by-one replacement argument, and then transitions into characteristic functions (CFs) as the main tool of Chapter 3.3.
1. Triangular Arrays and the Goal
Consider a triangular array of independent random variables:
\[\{X_{n,k}\}_{1 \le k \le n,\; n\ge 1},\]with:
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Mean zero
\(E[X_{n,k}] = 0.\) -
Unit total variance
\(\sum_{k=1}^n E[X_{n,k}^2] = 1.\)
This generalizes the iid CLT case, illustrated by the bottom example on page 1:
- Start with iid $x_k$ with mean 0 and variance 1.
- Form
\(X_{n,k} = \frac{x_k}{\sqrt n}, \qquad 1 \le k \le n.\)
Then
\(\sum_{k=1}^n E[X_{n,k}^2] = 1.\)
2. Need for the Lindeberg Condition
We “remove the iid assumption’’ by adding a single condition that controls large jumps:
Lindeberg Condition
For every $\varepsilon>0$, \(L_n(\varepsilon) = \sum_{k=1}^n E\!\left[X_{n,k}^2;\; \vert X_{n,k}\vert >\varepsilon\right] \;\xrightarrow[n\to\infty]{}\; 0. \tag{1}\)
The handwritten note on page 1 marks this as “Really need Lindeberg Condition” and references Theorem 3.4.5 (Lindeberg–Feller CLT).
Theorem (Lindeberg–Feller CLT)
Under independence, mean-zero, total variance 1, and the Lindeberg condition (1), the sum
\[S_n = \sum_{k=1}^n X_{n,k}\]converges in distribution:
\[S_n \Rightarrow Z\sim N(0,1).\]3. Lindeberg ⇒ No Dominant Term
(Page 1: Condition (A))
From Lindeberg’s condition we obtain:
\[\max_{1\le k\le n} E[X_{n,k}^2] \longrightarrow 0. \tag{2}\]Proof sketch (exactly as on page 1):
Fix $\varepsilon>0$. Let $k^*(n)$ be an index where the maximum variance occurs. Then:
\[E[X_{n,k^*}^2] = E[X_{n,k^*}^2;\, \vert X_{n,k^*}\vert >\varepsilon] + E[X_{n,k^*}^2;\, \vert X_{n,k^*}\vert \le\varepsilon].\]The first term is $\le L_n(\varepsilon)\to0$.
The second term is $\le \varepsilon^2$.
Since $\varepsilon$ is arbitrary, (2) follows.
Thus no single $X_{n,k}$ contributes macroscopic variance.
This is the triangular-array analogue of “maximal term goes to 0”.
4. CLT Proof via One-by-One Replacement
(Continuation of the argument from Lecture 34; Page 1 shows the same diagram.)
Let $Z_1, \dots, Z_n$ be iid $N(0,1)$. Define the comparison sum
\[Z_n^* = \sum_{k=1}^n \sigma_{n,k} Z_k, \qquad \sigma_{n,k}^2 := E[X_{n,k}^2].\]Then $Z_n^* \sim N(0,1)$.
Let
\[T_{n,k} = \sum_{m<k} X_{n,m} + \sum_{m>k} \sigma_{n,m} Z_m\](“before” + “after”; see the large diagram on page 1).
For $f\in C_B^3(\mathbb R)$ define:
\[I_{n,k} = \Big\vert E[f(T_{n,k}+X_{n,k})] - E[f(T_{n,k}+\sigma_{n,k} Z_k)] \Big\vert .\]Exactly as in Lecture 34:
\[\vert E[f(S_n)] - E[f(Z_n^*)]\vert \;\le\; \sum_{k=1}^n I_{n,k}. \tag{3}\]Taylor’s formula gives (page 1–2):
\[I_{n,k} \le C\, E\!\left( \vert X_{n,k}\vert ^2\,\mathbf{1}_{\vert X_{n,k}\vert >\varepsilon} + \vert X_{n,k}\vert ^3\,\mathbf{1}_{\vert X_{n,k}\vert \le\varepsilon} + \sigma_{n,k}^3\,E\vert Z_k\vert ^3 \right). \tag{4}\]- The first term is the Lindeberg term.
- The second term is $\le \varepsilon E[X_{n,k}^2]$.
- The third term: $\sigma_{n,k}^3 \le \sigma_{n,k}^2 = E[X_{n,k}^2]$.
Summing (page 1, end):
\[\sum_{k=1}^n I_{n,k} \le C L_n(\varepsilon) + \varepsilon. \tag{5}\]As $n\to\infty$, (1) makes the first term vanish; then let $\varepsilon\to0$.
Thus:
\[E[f(S_n)] - E[f(Z)] \to 0.\]So:
\[S_n \Rightarrow Z.\]5. Sufficient Condition for Lindeberg
(Page 1 bottom: “Example where Ln(ε) → 0”)
If for some $\delta>0$,
\[\sum_{k=1}^n E\big(\vert X_{n,k}\vert ^{2+\delta}\big) \;\xrightarrow[n\to\infty]{} 0, \tag{6}\]then the Lindeberg condition holds.
Proof from notes (page 1–bottom to page 2 top):
If $\vert X_{n,k}\vert >\varepsilon$, then $\vert X_{n,k}\vert ^{2+\delta} > \varepsilon^{2+\delta}$.
Thus:
Summing:
\[L_n(\varepsilon) \le \varepsilon^{-\delta} \sum_{k=1}^n E\big(\vert X_{n,k}\vert ^{2+\delta}\big) \to 0.\]So (6) is a convenient sufficient condition for CLT.
6. Beginning of Chapter 3.3 — Characteristic Functions
(Pages 2–3)
Your notes shift into CFs, with geometric diagrams showing the complex plane and the unit circle.
Definition (Characteristic Function)
For a random variable $X$:
\[\varphi_X(t)= E[e^{itX}] = E[\cos(tX)] + i\,E[\sin(tX)].\](Page 2 contains the unit circle diagram, labeled “complex plane”.)
Key identities (page 2)
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Magnitude:
$\vert a+ib\vert = \sqrt{a^2+b^2}$. -
Conjugate: \(\overline{\varphi(t)} = \varphi(-t).\)
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Basic properties (Theorem 3.3.1):
- $\varphi_X(0)=1$,
- $\vert \varphi_X(t)\vert \le 1$,
- $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ if $X\perp Y$.
Relation to the MGF
The moment generating function is
\[\psi_X(t) = E[e^{tX}].\]Comparison table written in notes (page 2):
- MGF uses $tX$ in the exponent.
- CF uses $itX$.
- CF exists for all real $t$; MGF may fail to exist.
Joint Characteristic Functions
For a vector $(X,Y)$,
\[\varphi_{(X,Y)}(s,t) = E[e^{i(sX + tY)}].\]Independence equivalently means:
\[X\perp Y \quad\Longleftrightarrow\quad \varphi_{(X,Y)}(s,t) = \varphi_X(s)\varphi_Y(t). \tag{7}\]The table on pages 2–3 works through a discrete example illustrating dependence vs independence using CFs.
Cheat–Sheet Summary — Lecture 35
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Lindeberg–Feller CLT generalizes the CLT to triangular arrays.
Condition:
\(L_n(\varepsilon) =\sum_k E[X_{n,k}^2; \vert X_{n,k}\vert >\varepsilon] \to 0.\) -
Lindeberg ⇒ no dominant term: $\max_k E[X_{n,k}^2]\to0$.
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One-by-one replacement (as in Lecture 34) yields: \(\vert E[f(S_n)] - E[f(Z)]\vert \le C L_n(\varepsilon) + \varepsilon.\)
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Sufficient condition:
$\sum_k E(\vert X_{n,k}\vert ^{2+\delta})\to0$ for some $\delta>0$. -
Characteristic functions introduced:
\(\varphi_X(t)=Ee^{itX},\quad \varphi_{X+Y}=\varphi_X\varphi_Y\text{ if independent.}\) -
CFs always exist, are bounded, and encode full distributional information.
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