Convergence in probability

Random variables $X_n$ converge in probability to $X$ whenfor every $\epsilon >0$.

Uniform convergence in probability means that the supremum of the absolute discrepancy on the entire index set tends to zero in probability. It controls evaluation at random indices in that set, unlike pointwise convergence in probability. On a fixed compact index set it is a special case of uniform convergence on compacts in probability.

A sequence of random variables $(X_n)$ is bounded in probability when ${\lim }_{K\to \infty }{\sup }_n\mathbb{P}(|X_n|>K)=0$. A sequence converging in probability to a finite random variable has this property, and its product with a sequence tending to zero in probability also tends to zero in probability.

The notation $Z_n=O_{\mathbb{P}}(a_n)$ means that $Z_n/a_n$ is bounded in probability asymptotically: for every $\epsilon >0$ some $C$ gives ${\mathrm{limsup}}_n\mathbb{P}(|Z_n|>C{a}_n)\le \epsilon$. A tail bound $\mathbb{P}(|Z_n|>z{a}_n)\le C_0/z^2$ is sufficient. In statements involving bandwidths, specify whether the constants must be uniform over bandwidth sequences.

A bound on ${\sup }_x\mathbb{P}(|Z_n(x)|>t)$ does not give the same bound on $\mathbb{P}({\sup }_x|Z_n(x)|>t)$. Even independent standard normal random variables each have uniformly bounded tails while their Gaussian maximum over $N$ indices grows at scale $\sqrt{\log N}$. This distinction matters in nonparametric statistics when a shrinking bandwidth permits many spatial windows.

If $X_n\stackrel{d}{\to }c$ for fixed real $c$, convergence of the distribution functions at $c-\epsilon$ and $c+\epsilon$ shows $\mathbb{P}(|X_n-c|>\epsilon )\to 0$. Thus convergence in distribution to a deterministic limit implies convergence in probability to that limit, even though the implication fails for nonconstant limits on a specified probability space.

For independent and identically distributed random variables with finite expected value, the sample mean converges in probability to that expected value.

A class $\mathcal{G}$ satisfies a uniform law of large numbers when its empirical means converge uniformly to their expectations:

A measurable-function class is pointwise separable if it has a countable subclass such that every member is the pointwise limit of a sequence from that subclass. With an integrable common envelope, dominated convergence theorem makes both empirical averages and population expectations converge along each such sequence. Thus the supremum of the absolute empirical errors equals the supremum over the countable subclass and is measurable. A finite integrable bracketing cover provides an integrable envelope by taking the maximum absolute value of all its endpoints.

A bracket $[\ell ,u]$ contains every measurable function $h$ satisfying $\ell \le h\le u$ pointwise. For a probability measure $P$, its $L^1(P)$ width is $P(u-\ell )$, with integrable endpoints. Finite bracketing at every positive width is a sufficient complexity condition for a uniform law of large numbers. Pointwise inequalities, or inequalities outside a single common null set, allow empirical inequalities to hold simultaneously throughout the class.

A measurable function $M$ is an integrable envelope of a function class for a class $\mathcal{H}$ under a probability law $P$ if $|h(x)|\le M(x)$ for every $h\in \mathcal{H}$ and $PM<\infty$. Such an envelope permits the dominated convergence theorem to control shrinking function brackets.

Suppose an integrable measurable-function class can be covered by finitely many function brackets of every positive $L^1(P)$ width. For an independent sample with common law $P$, the empirical measure satisfies ${\sup }_h|P_{n}h-Ph|\to 0$ on a common probability-one event. For one finite $\epsilon$-cover, the supremum is bounded by $\epsilon$ plus the largest empirical error among its endpoints. The strong law of large numbers makes that finite maximum vanish. Taking a countable sequence of widths decreasing to zero proves the assertion. A measurable supremum can be obtained from a pointwise separable function class; otherwise the probability-one-event formulation expresses the same pathwise conclusion.

If $\Theta$ is compact, $q(\theta ,x)$ is continuous in $\theta$ and measurable in $x$, and $\mathbb{E}{\sup }_{\Theta }|q(\theta ,X)|<\infty$, then an independent and identically distributed sample satisfies ${\sup }_{\Theta }|P_n{q}_{\theta }-P{q}_{\theta }|\to 0$ almost surely. Uniform continuity for each $x$, the dominated convergence theorem, and finite parameter nets produce arbitrarily narrow function brackets.

The empirical distribution function converges uniformly almost surely to the population distribution function.

If $X_n$ converges in probability to both $X$ and $Y$, then $X=Y$ almost surely. Indeed, the triangle inequality and a union bound show that $\mathbb{P}(|X-Y|>\epsilon )=0$ for every $\epsilon >0$.

Convergence in probability is equivalent toThe forward implication splits at a fixed error threshold; the reverse implication is Markov's inequality.

From $X_n\to X$ in probability, choose $n_k$ with
$\mathbb{P}(|X_{n_k}-X|>2^{-k})<2^{-k}$. The first Borel--Cantelli lemma then gives $X_{n_k}\to X$ almost surely.

Independent indicators with success probabilities $1/n$ converge to zero in probability, but successes occur infinitely often almost surely by the second Borel--Cantelli lemma.