Modes of Convergence: A.S., Probability, Lp, Distribution
"Xn converges to X" is four different theorems in this subject. This subchapter defines all four honestly, proves every implication arrow, and exhibits the example that shows none of the arrows can be reversed — the typewriter sequence, which every practitioner should know by name.
In this subchapter
7.1.1The four definitions
Let X₁, X₂, … and X be random variables on (Ω, F, P).
- Almost surely: P( Xn → X ) = 1 — the set of ω where the values converge has full measure. The strongest everyday mode; the a.s. language of 4.1 §4.
- In probability: for every ε > 0, P( |Xn − X| > ε ) → 0 — deviations of any fixed size become unlikely. No single ω needs to behave; the population of deviants thins out.
- In Lp (p ≥ 1): E|Xn − X|p → 0. For p = 2: mean-square convergence, the metric behind least squares.
- In distribution: P(Xn ≤ t) → P(X ≤ t) at every continuity point t of the limit's CDF. A statement about laws, not variables: 4.2's CDFs now doing asymptotic duty.
Uniform variables with a shrinking spike: Xn = n·1(0,1/n] on ([0,1], ℒ, λ) — the moving bump of 5.2 Example 1. To 0: pointwise at every ω (eventually 0), so a.s. — yes. In probability: P(Xn > ε) = 1/n → 0 — yes. In L¹: E|Xn| = 1 ↛ 0 — no. In distribution: P(Xn ≤ t) → P(0 ≤ t) for every fixed t ≠ 0 (check: at t > 0, eventually n > t so P = 1; limit equals the point mass at 0) — yes, to the Dirac law δ₀. One sequence, three yeses and one no: the modes are genuinely different verdicts on the same object.
7.1.2The hierarchy, with proofs of every arrow
Almost sure ⇒ in probability ⇒ in distribution; Lp ⇒ in probability (any p ≥ 1). Moreover if Xn → X a.s. and |Xn| ≤ Z with Z ∈ Lp, then also Lp — the dominated convergence theorem of 5.2 in convergence clothing.
A.s. ⇒ probability. Fix ε. The event { |Xn − X| > ε i.o. } is contained in the complement of the convergence set — null. By Borel–Cantelli I (converse direction): if P(|Xn−X| > ε) did not tend to 0 along a subsequence... the clean route: the events Bn = ⋃k≥n{|Xk − X| > ε} decrease, and their intersection is contained in {not converging} (any ω converging is eventually within ε), so lim P(Bn) = P(⋂Bn) = 0 by continuity from above (2.2 Theorem 3, the finite-first-measure case — the a.s. hypothesis enters as null-intersection). Since P(|Xn−X| > ε) ≤ P(Bn), done.
Lp ⇒ probability. For any ε: P(|Y| > ε) ≤ E|Y|p/εp with Y = Xn − X — Markov's inequality, proved in one line: E|Y|p ≥ E[|Y|p1{|Y|>ε}] ≥ εpP(|Y|>ε) (a monotonicity + indicator split, 5.1 machinery). Apply with Y = Xn−X: the right side → 0.
Probability ⇒ distribution. At a continuity point t of the limit CDF and any s > t: P(Xn ≤ t) ≤ P(X ≤ s) + P(|Xn−X| > s−t) (triangle through s), and symmetrically with t, s swapped: sandwich P(X ≤ s)-type bounds, let n → ∞ then s ↓ t — continuity of the limit CDF at t makes the two ends meet at P(X ≤ t). ∎
7.1.3The typewriter: why no arrow reverses
On ([0,1], ℒ, λ), enumerate dyadic intervals one sweep at a time: X₁ = 1[0,1]; then X₂ = 1[0,1/2], X₃ = 1[1/2,1]; then four intervals of length 1/4, then eight of length 1/8, and so on — a cursor typing across the unit interval, ever finer. In probability to 0: for ε ∈ (0,1), P(Xn > ε) is the current sweep's interval length, 2−k → 0. Not a.s.: every fixed ω belongs to one interval of every sweep, so Xn(ω) equals 1 infinitely often and 0 infinitely often — the sequence oscillates forever and never converges, at every ω. But L¹ holds too: with height 1, E|Xn| = 2−k on the k-th sweep, which tends to 0 — so the plain typewriter does not separate L¹ from in-probability. The scaling that breaks L¹ while keeping in-probability is height 2k on the k-th sweep: then P(Xn > ε) = 2−k → 0 still, but E|Xn| = 2k·2−k = 1 ↛ 0. So: the plain typewriter separates a.s. from in-probability; the scaled typewriter (height 2k) separates L¹ from in-probability; and Example 1's spike separates a.s. from L¹. Three examples, three broken arrows — as promised, no implication reverses.
For the distribution arrow: take Xn independent, all with the same law as X. Then Xn → X in distribution trivially (identical CDFs) — while P(|Xn − X| > ε) does not vanish in general: e.g. all uniform on [0,1] independent, P(|Xn − X| > 1/2) = 1/2 forever. Convergence in distribution says the laws converge; the variables need not be connected by any coupling at all — the mode that a spreadsheet of simulations silently assumes is the weakest of the four.
7.1.4Convergence in distribution: the odd one out
Three structural facts that make the fourth mode different. (1) The variables Xn and X need not live on the same space — the definition mentions only CDFs, so the mode transfers across models; this is exactly what lets a CLT proved once apply to every problem that fits. (2) The "continuity points" clause is not pedantry: Xn ≡ 1/n → 0 but P(Xn ≤ 0) = 0 ↛ 1 = P(X ≤ 0) — at the atom of the limit, the CDFs jump over each other, so convergence is required only where the limit is continuous; that this clause is exactly right (no more, no less) is the content of the Portmanteau theorem, a checklist of equivalent formulations we state in 7.3 where it earns its keep. (3) Continuous mapping: if Xn → X in distribution and g is continuous, then g(Xn) → g(X) in distribution — one δ-management line from the sandwich of Theorem 1's third part; this is the license that turns one CLT into an unlimited supply (apply to ln, exp, payoffs).
7.1.5Cauchy in probability: the completeness surprise
A sequence convergent in probability in the Cauchy sense — P(|Xn − Xm| > ε) → 0 as n, m → ∞ — converges in probability to some X, even though no candidate limit was given in advance. So the in-probability world is complete: every Cauchy sequence has a limit. The proof runs by extracting a fast subsequence (a.s.-convergent by Borel–Cantelli I: choose nk with the pairwise deviations summably small), declaring its limit as X, and upgrading back to the full sequence — a genuinely beautiful argument, and the blueprint for 7.2's strong law. Contrast: the a.s. mode is not "complete" in any naive sense without the same subsequence technology — and Lp completeness is the Riesz theorem (we flag it, and use it in 8.3 for martingale L² theory).
7.1.6Which mode does your problem need?
- Pathwise guarantees ("this strategy's drawdown eventually ends" — a statement about the realized path): you need a.s. In-probability is not enough — the typewriter's deviants visit every ω.
- Risk at a horizon ("P(loss > VaR) is under control at time n"): in probability is exactly the right mode — the statement is a single-time probability.
- Mean-square and variance statements (estimator efficiency, L² theory,Kalman filters): L² convergence, and the hypotheses (moments!) must actually hold — 5.5's tail checklist first.
- Approximate distributions ("the average is approximately normal, so price with the normal table"): in distribution — the CLT's mode (7.3), sufficient exactly when downstream computations use only law-level information.
The discipline: name the mode before trusting the conclusion. "Converges" without a mode is the applied-probability equivalent of "at random" without a measure — 6.4's lesson, wearing asymptotic clothes.
Four modes, one hierarchy, no reversals: every convergence claim is four claims in one, and only the named one is proved.
The modes are in place. The next subchapter proves the granddaddy of all convergence results — the laws of large numbers: Chebyshev's weak law in a few lines, the strong law with the full subsequence/Borel–Cantelli proof (under a fourth-moment hypothesis, honestly flagged), and the Cauchy hypothesis failure showing that "averages converge" is a theorem with conditions, not a law of nature.