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A sequence of random variables can converge almost surely while its expectations refuse to converge. The missing control is often in the tails: tiny probabilities may carry increasingly large values. Uniform integrability rules out that escaping mass uniformly across the family. In martingale theory it marks the difference between merely reaching a finite almost-sure limit and converging to that limit in L¹ with expectations preserved.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Separate three kinds of convergence
Almost-sure convergence follows individual sample paths. Convergence in probability controls the probability of a visible error. L¹ convergence controls the expected absolute error E|Xn−X|.
L¹ convergence implies convergence in probability, but almost-sure convergence alone does not control averages. A rare value can become larger as it becomes less likely.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Define uniform integrability through tail control
A family {Xn} is uniformly integrable when supn E[|Xn| 1{|Xn|>K}] tends to zero as K tends to infinity. One cutoff K must make every tail expectation small.
Each variable being integrable is not enough. Uniformity prevents the index n from moving more expected mass beyond every fixed cutoff.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work the escaping-mass counterexample
Let Xn=n when an event An of probability 1/n occurs, and let Xn=0 otherwise. If the events are nested as An=(0,1/n) on a uniform sample space, then Xn converges almost surely to zero.
| Quantity | Calculation | Result |
|---|---|---|
| P(Xn≠0) | 1/n | Tends to 0 |
| E[Xn] | n×(1/n) | 1 for every n |
| Almost-sure limit | 0 | Every positive sample point eventually leaves An |
| L¹ error to 0 | E|Xn| | 1 for every n |
The sequence is L¹-bounded because supn E|Xn|=1, yet it is not uniformly integrable. For any fixed K, choose n>K; then the whole expectation lies beyond K.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Contrast an actually controlled family
Now let Yn=n only on an event of probability 1/n², and zero otherwise. Then E|Yn|=1/n and Yn converges almost surely to zero under the same nested construction.
| Family | Spike size | Spike probability | Expected tail mass |
|---|---|---|---|
| Xn | n | 1/n | 1 |
| Yn | n | 1/n² | 1/n |
| Bounded Zn | At most M | Any | 0 once K>M |
For Yn, the largest remaining tail expectation beyond K is at most about 1/K. The tail mass disappears uniformly, so the family is uniformly integrable.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Use the convergence upgrade correctly
If Xn converges in probability to X and {Xn} is uniformly integrable, then X is integrable and Xn converges to X in L¹. Conversely, L¹ convergence makes the family uniformly integrable.
This is not a licence to replace every dominated-convergence check with the phrase “bounded expectation”. A single integrable dominating variable is sufficient for uniform integrability, but a common L¹ bound alone is not.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Connect martingale limits with closed martingales
Many nonnegative or L¹-bounded supermartingales converge almost surely under Doob-type convergence results. The limit need not preserve the starting expectation.
A uniformly integrable martingale has a stronger representation: there is an integrable terminal variable X∞ such that Xn=E[X∞|Fn], and Xn→X∞ in L¹. This is sometimes called a closed martingale.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: stopping a fair game
A stopped martingale can be well behaved for every bounded stopping time while an unbounded stopping rule exposes rare large losses or gains. Optional-stopping conclusions require conditions such as bounded stopping, controlled increments or uniform integrability of the stopped family.
The transfer check is not “the game is fair at each step”. It is whether the collection {Mτ∧n} keeps its tails under uniform control as n grows.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the existing probability route
The guide to Borel–Cantelli lemmas and infinitely-often events helps distinguish pathwise recurrence from expectation control. The guide to exchangeability and de Finetti mixtures shows a different way hidden structure changes unconditional behaviour.
Use first success, retry limits and unfinished runs when the main issue is truncation of attempts rather than convergence of random variables.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Uniform integrability does not mean the variables are uniformly bounded, independent or identically distributed. It controls expected tail mass, not the pointwise maximum.
Almost-sure convergence plus supn E|Xn|<∞ is still insufficient for L¹ convergence. For p>1, a uniform Lp bound is sufficient by a tail estimate, but p=1 is exactly where the rare-spike counterexample lives.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical proof and evidence routine
State the convergence mode, identify the proposed limit and inspect the tails separately from the central region. Test whether a dominating variable, an Lp bound with p>1, de la Vallée-Poussin growth or a direct tail calculation proves uniform integrability.
For a martingale or stopped process, record the filtration, integrability at each time, stopping rule and the family whose uniform integrability is actually needed. Do not infer expectation preservation from pathwise convergence alone.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does uniform integrability imply bounded expectations?
Yes. Split each expectation below and above one sufficiently large cutoff.
Do bounded expectations imply uniform integrability?
No. The rare-spike family Xn has expectation one for every n but fails uniform integrability.
Why is L¹ convergence useful?
It controls expected absolute error and guarantees convergence of expectations under the stated integrability conditions.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Stanford’s Statistics 310A probability notes develop uniformly integrable martingales and the L¹ convergence theorem. Berkeley’s Probability Theory lecture notes connect martingale convergence with uniform integrability. The rare-spike, controlled-spike and stopped-family checks here are original teaching examples.
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