Lecture 39 — Poisson Convergence & Domains of Attraction
We study two distinct but related limit behaviors:
- Poisson convergence (Law of Rare Events),
- Lindeberg–Feller CLT and truncation,
- Lévy’s criterion for domain of attraction of the normal law,
- The Berry–Esseen quantitative CLT.
1. A CLT Example: Harmonic Means of Rare Events
(Page 1 of notes.)
Let ${Y_k}_{k\ge1}$ be independent with:
\[P(Y_k = 1) = \frac{1}{k}, \qquad P(Y_k = 0) = 1-\frac{1}{k}.\]Then:
\[S_n = \sum_{k=1}^n Y_k.\]Compute expectation:
\[E[S_n] = \sum_{k=1}^n \frac{1}{k} \sim \log n.\]Variance:
\[\mathrm{Var}(S_n) = \sum_{k=1}^n \left(\frac{1}{k}-\frac{1}{k^2}\right) \sim \log n.\]Thus the natural normalization is:
\[\frac{S_n - \log n}{\sqrt{\log n}} \Rightarrow N(0,1).\]Writing as a triangular array
Define:
\[X_{n,k} = \frac{Y_k - \frac{1}{k}}{\sqrt{\log n}}, \qquad 1\le k\le n.\]Then:
\[S_n - \log n = \sqrt{\log n}\sum_{k=1}^n X_{n,k}.\]The Lindeberg condition holds trivially because:
\[\vert Y_k - 1/k\vert \le 1, \qquad \vert X_{n,k}\vert \le \frac{1}{\sqrt{\log n}}\to0.\]Thus $\max_{1\le k\le n}\vert X_{n,k}\vert \to0$, which implies Lindeberg.
So the Lindeberg–Feller CLT applies and yields the above normal limit.
2. Truncation Example for Arrays with Infinite Moments
(Page 1 → page 2.)
Consider iid ${X_{n,m}}_{m=1}^n$ for each fixed $n$, but with a heavy-tailed distribution that changes with $n$. Suppose:
\[P\left(X_{n,1} = \pm \frac{1}{\sqrt n}\right) = \frac12 - \frac{1}{2n^2},\]and with very small probability:
\[P\left(X_{n,1} = \pm \frac{4^k}{\sqrt n}\right) = \frac{1}{2n^2 2^k},\qquad k=1,2,3,\dots\]You can see on page 1–2 that:
\[\sum_{k=1}^\infty \frac{1}{2^k} = 1,\]so this defines a probability distribution.
But:
\[E\vert X_{n,1}\vert = \infty, \qquad E[X_{n,1}^2] = \infty,\]by the diverging tail contributions (page 2).
Thus we cannot apply the CLT directly.
2.1 Define truncation
Define the “good” variables:
\[Y_{n,m} = X_{n,m}\mathbf{1}_{\vert X_{n,m}\vert \le 1/\sqrt n}.\]Then from page 2:
\[Y_{n,1} = \begin{cases} 0, & \text{with prob } 1/n \pm\,1/\sqrt n, &\text{with prob } \frac12 - \frac{1}{2n^2}. \end{cases}\]Compute:
\[E[Y_{n,1}] = 0,\] \[E[Y_{n,1}^2] = \frac{1}{n}\Big(\frac{1}{2}-\frac{1}{2n^2}\Big) = \frac{1}{n} - \frac{1}{n^3}.\]Thus:
\[\sum_{m=1}^n \mathrm{Var}(Y_{n,m}) = 1 - \frac{1}{n^2} \to 1.\]So the triangular array ${Y_{n,m}}$ satisfies Lindeberg and hence:
\[T_n = \sum_{m=1}^n Y_{n,m} \Rightarrow N(0,1).\]2.2 Compare $T_n$ and $S_n = \sum_{m=1}^n X_{n,m}$
Page 2–3:
\[P(S_n \neq T_n) \le n\,P(\vert X_{n,1}\vert >1/\sqrt n) = \frac{n}{n^2} = \frac1n \to 0.\]Thus:
\[S_n - T_n \xrightarrow{P} 0.\]By Slutsky:
\[\boxed{ S_n \Rightarrow N(0,1). }\]This is the same structure as in Lecture 38: truncation recovers CLT even when raw moments diverge.
3. Lévy’s Domain of Attraction Criterion
(Page 3.)
We say iid $X_1,X_2,\dots$ are in the domain of attraction of the normal law if there exist constants $a_n\in\mathbb R$, $b_n>0$ such that:
\[\frac{S_n - a_n}{b_n} \Rightarrow N(0,1).\]Paul Lévy’s condition:
\[\boxed{ \frac{y^2 P(\vert X\vert >y)} {E[X^2;\,\vert X\vert \le y]} \;\xrightarrow[y\to\infty]{}\; 0. } \tag{Lévy}\]This is both necessary and sufficient for the normal domain of attraction.
Your notes comment:
“If does not hold, forget about truncation.”
i.e., no normalization (even nonlinear) will yield a CLT-like limit without severe modifications.
4. Berry–Esseen Theorem
(Page 3.)
If $X_1,X_2,\dots$ iid with:
\[E[X]=0, \quad \mathrm{Var}(X)=1,\quad E\vert X\vert ^3<\infty,\]then:
\[\vert F_n(x) - \Phi(x) \vert \le \frac{3E\vert X\vert ^3}{\sqrt n}, \qquad \forall x\in\mathbb R,\]where:
\[F_n(x)=P\left(\frac{S_n}{\sqrt n}\le x\right), \qquad \Phi(x)=P(Z\le x),\ Z\sim N(0,1).\]Notes emphasize:
- “Convergence is uniform, not pointwise.”
- $\sup \vert F_n-\Phi\vert \sim 1/\sqrt n$.
5. Poisson Convergence (Law of Rare Events)
(Page 3–4.)
If:
\[Y \sim \mathrm{Poisson}(\lambda),\qquad P(Y=k) = e^{-\lambda}\frac{\lambda^k}{k!},\]then:
Classical result:
\(\mathrm{Bin}(n, \lambda/n) \Rightarrow \mathrm{Poisson}(\lambda).\)
5.1 General Poisson Convergence Theorem
(Page 4.)
Let ${X_{n,m}}_{1\le m\le n}$ be independent Bernoulli variables:
\[P(X_{n,m}=1) = p_{n,m}, \qquad P(X_{n,m}=0) = 1 - p_{n,m}.\]Assume:
- \[\sum_{m=1}^n p_{n,m} \to \lambda\ge0.\]
- \[\max_{1\le m\le n} p_{n,m} \to 0.\]
Then:
\[S_n = \sum_{m=1}^n X_{n,m} \Rightarrow \mathrm{Poisson}(\lambda).\]This is the Poisson limit for triangular arrays.
5.2 Proof via characteristic functions
CF of Bernoulli:
\[\varphi_{X_{n,m}}(t) = 1 + p_{n,m}(e^{it}-1).\]Thus:
\[\varphi_{S_n}(t) = \prod_{m=1}^n \big(1 + p_{n,m}(e^{it}-1)\big).\]Set:
\[a_{n,m}=p_{n,m}(e^{it}-1).\]If conditions (1)–(3) of the triangular-array lemma hold:
- $\sum_{m=1}^n a_{n,m} \to \lambda(e^{it}-1)$,
- $\sup_n \sum_m \vert a_{n,m}\vert <\infty$,
- $\max_m \vert a_{n,m}\vert \to 0$,
then:
\[\varphi_{S_n}(t) \to \exp\left(\lambda(e^{it}-1)\right),\]the CF of Poisson$(\lambda)$.
Thus:
\[S_n \Rightarrow \mathrm{Poisson}(\lambda).\]Cheat–Sheet Summary — Lecture 39
-
Rare-event sum $S_n=\sum Y_k$ with $P(Y_k=1)=1/k$ satisfies a CLT: \(\frac{S_n-\log n}{\sqrt{\log n}} \Rightarrow N(0,1).\)
-
Heavy-tailed triangular arrays may require truncation to satisfy Lindeberg.
If $S_n-T_n\to0$ in probability and $T_n\Rightarrow N(0,1)$, then $S_n\Rightarrow N(0,1)$. -
Lévy’s criterion:
\(\frac{y^2 P(\vert X\vert >y)}{E[X^2;\vert X\vert \le y]}\to 0 \iff X\ \text{in Gaussian DOA}.\) -
Berry–Esseen gives uniform CLT rate
$\sup\vert F_n-\Phi\vert \le 3E\vert X\vert ^3/\sqrt n$. -
Poisson convergence theorem:
If $\sum p_{n,m}\to\lambda$ and $\max p_{n,m}\to0$, then
\(S_n=\sum X_{n,m} \Rightarrow \mathrm{Poisson}(\lambda).\)
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