Edit this page on GitHub

Lecture 25 — SLLN Completion, Weighted Triangular Arrays, Random Series

1. Completing the SLLN Proof

We restate the goal:

Let ${X_k}_{k\ge1}$ be iid with
\(E\vert X\vert < \infty, \qquad E[X] = \mu.\) Define \(S_n = \sum_{k=1}^n X_k.\) Then \(\frac{S_n}{n} \xrightarrow{a.s.} \mu.\)

Recap of the steps

Step 1: Truncation

Define
\(Y_k = X_k \mathbf{1}_{\{\vert X_k\vert \le k\}}.\)

Since
\(\sum_{k=1}^\infty P(\vert X_k\vert > k) = \sum_{k=1}^\infty P(\vert X\vert >k) \le E\vert X\vert < \infty,\) Borel–Cantelli I gives
\(P(X_k \ne Y_k\ \text{i.o.}) = 0.\)

Therefore, for all sufficiently large $k$, $Y_k = X_k$ a.s.

Let
\(T_n = \sum_{k=1}^n Y_k.\)

If we prove that $T_n/n \to \mu$, the same holds for $S_n/n$.

Step 2: Second-moment summability

We showed last time:

\[\sum_{k=1}^\infty \frac{E[Y_k^2]}{k^2} < \infty.\]

This is essential for controlling block variances.

Step 3: Block decomposition

Define “block increments” for $n\ge1$:

\[U_n = \sum_{k=2^n+1}^{2^{n+1}} \frac{Y_k - E[Y_k]}{2^n}.\]

Then (as on page 1 of the PDF):

\[\sum_{n=1}^\infty E[U_n^2] = \sum_{n=1}^\infty \sum_{k=2^n+1}^{2^{n+1}} \frac{\mathrm{Var}(Y_k)}{(2^n)^2} \le 4 \sum_{k=1}^\infty \frac{E[Y_k^2]}{k^2} < \infty.\]

Hence, by Borel–Cantelli or the Kolmogorov square-summability lemma:

\[U_n \xrightarrow{a.s.} 0.\]

Consequently,

\[\frac{T_{2^n} - E[T_{2^n}]}{2^n} = \sum_{m=1}^{n-1} \frac{2^m}{2^n} U_m \xrightarrow{a.s.} 0.\]

(Since $2^m/2^n = 2^{m-n}\to 0$ for fixed $m$, and the series is a.s. summable.)

Step 4: Compute the limit of the means

Because $Y_k = X_k \mathbf{1}_{{\vert X_k\vert \le k}}$,

\[E[Y_k] = E(X; \vert X\vert \le k) \xrightarrow{k\to\infty} E[X] = \mu.\]

Then

\[\frac{E[T_{2^n}]}{2^n} = \frac{1}{2^n} \sum_{k=1}^{2^n} E[Y_k] \xrightarrow{n\to\infty} \mu.\]

Combining the two limits:

\[\frac{T_{2^n}}{2^n} = \frac{E[T_{2^n}]}{2^n} + \frac{T_{2^n}-E[T_{2^n}]}{2^n} \xrightarrow{a.s.} \mu.\]

Step 5: Extend to all $n$

For any $2^n \le m \le 2^{n+1}$,

\[\frac{T_{2^n}}{2^{n+1}} \le \frac{T_m}{m} \le \frac{T_{2^{n+1}}}{2^n}.\]

Since

\[\frac{T_{2^n}}{2^n} \to \mu, \qquad \frac{T_{2^{n+1}}}{2^{n+1}} \to \mu,\]

we squeeze the middle term and conclude:

\[\boxed{ \frac{T_m}{m} \to \mu \quad a.s. }\]

Finally, since $S_m = T_m$ for all large $m$ a.s.,

\[\boxed{ \frac{S_m}{m} \xrightarrow{a.s.} \mu. }\]

2. Weighted Convergence of Triangular Arrays

Your notes introduce a general weighted convergence lemma (yellow text, page 1):

Let $a_{n,k}\ge 0$ be a triangular array of weights, with

  1. $a_{n,k}\ge0$,
  2. $\sum_{k=1}^n a_{n,k} = 1$,
  3. $a_{n,k} \to 0$ for each fixed $k$.

Let $b_k \to b$. Then

\[\sum_{k=1}^n a_{n,k} b_k \xrightarrow{n\to\infty} b.\]

Idea: These arrays behave like approximate identities.
This lemma is implicitly used in the argument converting block-convergence into full convergence.

3. Convergence of Random Series

Tail σ–fields

Given ${X_k}$, define the tail σ–field:

\[\mathcal{F}_{[n,\infty)} = \sigma(X_n, X_{n+1}, \dots).\]

Then

\[\mathcal{T} = \bigcap_{n=1}^\infty \mathcal{F}_{[n,\infty)}\]

is the tail σ–field.

Characterization of convergence of ∑X_k

The event

\[\{S_n \text{ converges}\} = \left\{ \forall \epsilon>0\ \exists N\ \text{such that } \vert S_l-S_m\vert <\epsilon\ \text{ for all } l>m\ge N \right\}\]

is a tail event: it depends only on ${X_k:k\ge N}$.

Hence it belongs to $\mathcal{T}$.

When the $X_k$ are iid, Kolmogorov’s 0–1 law applies:
the event of convergence of a random series has probability 0 or 1.

Example 1 (page 2)

Let $A_n$ be events such that

\[X_n = \mathbf{1}_{A_n}.\]

Then

\[\sum X_n \text{ converges} \quad\Longleftrightarrow\quad \mathbf{1}_{A_n} \text{ occurs only finitely often}.\]

Thus this reduces to Borel–Cantelli.

Example 2

If $\sum E\vert X_n\vert <\infty$ and the $X_n$ are independent, then

\[\sum X_n \quad\text{converges a.s.}\]

This is the classical Kolmogorov convergence criterion for random series.

Cheat-Sheet Summary — Lecture 25

  • The SLLN is obtained by:
    1. Truncation at level $k$: $Y_k = X_k1_{{\vert X\vert \le k}}$.
    2. BC I ensures truncation differs only finitely often.
    3. Prove $\sum E[Y_k^2]/k^2<\infty$.
    4. Use block decomposition $U_n$ to show $T_{2^n}/2^n \to \mu$.
    5. Squeeze intermediate values to conclude $S_n/n \to \mu$ a.s.

  • Weighted triangular array lemma:
    If weights sum to 1 and vanish on fixed indices, they preserve limits.

  • Convergence of random series is a tail event, hence 0–1 when iid.
Edit this page on GitHub

Comments