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Lecture 27 — A.s. Convergence via Variance Summability & Kolmogorov’s 3–Series Theorem

This lecture contains:

Kolmogorov’s convergence criteria via variance summability

A uniform maximal bound implying Cauchy convergence almost surely

Kolmogorov’s Three–Series Theorem (full statement)

Interpretation and proof sketch

1. Convergence of ∑ Xₙ under Variance Summability

Let ${X_n}_{n\ge1}$ be independent. Write

\[S_n = \sum_{k=1}^n X_k.\]

Theorem (Durrett 2.5.3)

Assume $E[X_n]=0$.

If
$\sum_{n=1}^\infty E(X_n^2) < \infty,$ then
$\sum_{n=1}^\infty X_n \quad \text{converges a.s.}$
If
$\sum_{n=1}^\infty \operatorname{Var}(X_n) < \infty,$ then
$\sum_{n=1}^\infty \bigl(X_n - E[X_n]\bigr) \quad\text{converges a.s.}$

Remark (From the note on page 1)

No moment assumptions beyond finite variance are needed.
If $\sum E[X_n^2]<\infty$, then automatically $E\bigl(\sum X_n^2\bigr) < \infty$ and

\[\sum X_n^2 < \infty \quad \text{a.s.}\]

2. Proof of Statement (1)

We show that ${S_n}$ is a Cauchy sequence almost surely.

Let $M<N$ and consider:

\[S_N - S_M = \sum_{k=M+1}^N X_k.\]

By independence and Chebyshev:

\[P\!\left( \max_{M\le m\le N} \vert S_m - S_M\vert > \epsilon \right) \;\le\; \frac{\operatorname{Var}(S_N - S_M)}{\epsilon^2} = \frac{\sum_{k=M+1}^N \operatorname{Var}(X_k)}{\epsilon^2}.\]

Letting $N\to\infty$:

\[P\!\left( \sup_{m\ge M} \vert S_m - S_M\vert > \epsilon \right) \le \frac{\sum_{k=M+1}^\infty \operatorname{Var}(X_k)}{\epsilon^2} \;\xrightarrow[M\to\infty]{}\; 0,\]

because $\sum\operatorname{Var}(X_k)$ converges.

Now define the nonnegative random variables:

\[W_M = \sup_{m\ge M} \vert S_m - S_M\vert .\]

The sequence $W_M$ is monotone decreasing.
We have $W_M \xrightarrow{P} 0$.
A monotone sequence converging in probability must converge almost surely.

Hence:

\[W_M \downarrow W,\qquad W=0 \ \text{a.s.}\]

\[\sup_{n,m\ge M}\vert S_n - S_m\vert \;\xrightarrow{M\to\infty}{a.s.}\; 0.\]

Thus ${S_n}$ is a.s. Cauchy, hence convergent almost surely.

3. Kolmogorov’s Three–Series Theorem

This is one of the most important convergence theorems for independent random variables.

Let ${X_k}_{k\ge1}$ be independent.
Fix a truncation constant $A>0$, and define:

\[Y_k = X_k \, \mathbf{1}_{\{\vert X_k\vert \le A\}}, \qquad M_k = E[Y_k].\]

The theorem gives necessary and sufficient conditions for the random series

$\sum_{k=1}^\infty X_k$ to converge almost surely.

Theorem (Kolmogorov’s Three–Series Theorem)

The series $\sum_{k=1}^\infty X_k$ converges a.s. iff all three series below converge:

Large jumps are rare: $\sum_{k=1}^\infty P(\vert X_k\vert > A) < \infty.$
The expected truncated values sum: $\sum_{k=1}^\infty M_k \quad\text{converges}.$
The truncated variances sum: $\sum_{k=1}^\infty \operatorname{Var}(Y_k) < \infty.$

These conditions do not depend on the choice of $A>0$.

4. Interpretation (from diagrams and notes on pages 1–2)

Condition (1): “Rare large jumps”

The event $\vert X_k\vert >A$ must occur only finitely many times (Borel–Cantelli II).
This ensures the sequence behaves like the truncated version except for finitely many terms.

Condition (2): “Truncated drift converges”

Since the truncated series differs from the original series only on a finite set (from condition (1)), the sum of expectations must converge for the overall drift not to diverge.

Condition (3): “Summable fluctuations”

This matches Theorem 2.5.3: if the variances of the truncated increments sum, then the truncated centered series

\[\sum (Y_k - M_k)\]

converges almost surely.

5. Proof Sketch (matching notes on pages 1–2)

Assume conditions (1)–(3).

(3) implies
$\sum_{k=1}^\infty (Y_k - M_k) \quad\text{converges a.s.}$ by the variance summability theorem proved above.
(2) implies
$\sum_{k=1}^\infty M_k \quad\text{converges},$ a deterministic series.
(1) implies via Borel–Cantelli II that $\vert X_k\vert >A$ occurs only finitely often.
Thus: $X_k = Y_k \quad\text{for all but finitely many }k.$