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A sequence {Xₙ} converges to X in probability if for all ε > 0, lim_{n→∞} P(|Xₙ - X| > ε) = 0. Intuitively, Xₙ is close to X with high probability for large n. Convergence in probability is weaker than almost sure convergence but stronger than convergence in distribution.
In deterministic analysis, a sequence of numbers xₙ converges to L if xₙ gets arbitrarily close to L for large enough n — every number in the tail of the sequence eventually lands near L. For random variables, the situation is richer: Xₙ is not a single number but a whole distribution. What does it mean for a random variable to "converge"? There are several answers depending on what you require. Convergence in probability is the most commonly encountered notion, and it has a natural intuitive reading.
The formal definition says: Xₙ converges to X in probability if, for every tolerance ε > 0, the probability that Xₙ is more than ε away from X goes to zero as n → ∞. In notation: P(|Xₙ − X| > ε) → 0 for all ε > 0. Concretely, pick any small margin — say, ε = 0.01. For large enough n, the chance that Xₙ differs from X by more than 0.01 becomes negligible. It's not that Xₙ is guaranteed to be close to X; it's that *most* of the probability mass of Xₙ is concentrated near X, and the exceptional events (large deviations) become rarer and rarer.
Think of a shrinking distribution as the key image. If Xₙ has a normal distribution with mean 0 and variance 1/n, then as n → ∞, the distribution collapses to a spike at 0. For any ε, the probability of landing outside (−ε, ε) is the tail probability of N(0, 1/n), which goes to 0. So Xₙ → 0 in probability. Notice that no individual outcome is guaranteed to be close to 0 — the randomness doesn't disappear, but the mass of the distribution concentrates. This is different from saying "Xₙ always stays near 0."
This distinction matters when comparing convergence modes. Almost sure convergence requires that the set of outcomes where Xₙ does *not* converge to X has probability zero — every path (except a null set) eventually stays near X. Convergence in probability is weaker: it only requires that the *probability* of straying far from X vanishes, not that every path behaves well. A classic counterexample (the "typewriter sequence") shows that convergence in probability does not imply almost sure convergence. Convergence in probability is the mode relevant to the Weak Law of Large Numbers: the sample mean converges in probability to the true mean, even though individual samples may occasionally be far off.