For ordinary numbers, "x_n \to x" means one thing. For a sequence of random variables (X_n) there are several genuinely different senses in which it can converge to a limit X — and they are not equivalent. Pinning down which mode you mean is essential, because the limits that define the Itô integral, the law of large numbers, and the central limit theorem each use a different one.

The four modes

• Almost surely (X_n \xrightarrow{a.s.} X): the paths converge for every outcome except a set of probability zero — \mathbb{P}\!\left(\lim_{n\to\infty} X_n = X\right) = 1.
• In probability (X_n \xrightarrow{\mathbb{P}} X): large gaps become rare — \mathbb{P}\!\left(|X_n - X| > \varepsilon\right) \to 0 \quad\text{for every } \varepsilon > 0.
• In L^p (mean of order p; p = 2 is mean-square): the average error shrinks — \mathbb{E}\!\left[|X_n - X|^p\right] \to 0.
• In distribution (X_n \xrightarrow{d} X): only the shape converges — F_{X_n}(x) \to F_X(x) at every point x where F_X is continuous. (The continuity proviso matters: at a jump of F_X the CDFs need not agree.)

The implication hierarchy

The modes line up in a strict order. Step through the diagram: both almost-sure and L^p convergence imply convergence in probability, which in turn implies convergence in distribution. None of the arrows reverse, and there is no arrow between a.s. and L^p in either direction.

Why this matters here

The Itô integral is built as an L^2 (mean-square) limit of integrals of simple processes — mean-square convergence is exactly the notion under which that construction closes up, courtesy of the Itô isometry. Meanwhile the weak law of large numbers is a statement of convergence in probability, the strong law is almost-sure convergence, and the central limit theorem is convergence in distribution. Knowing which mode is in play tells you exactly what each theorem does, and does not, promise.