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Convergence in probability is metrized by $d_{0}$

Statement

Durrett's exercise gives the equivalent bounded-transform metric with integrand $t / (1 + t)$ ; the proof below establishes the $\min (1, t)$ variant.

The formula $d_{0} ([X], [Y]) = E [\min (1, ∣ X - Y ∣)]$ is a metric on real random variables modulo almost-sure equality. Moreover, $d_{0} ([X_{n}], [X]) \to 0 ⟺ X_{n} \to X in probability .$

Facts & Assumptions

Given: Real random variables $X, Y, Z$ , and a sequence $(X_{n})$ , on one probability space.

[L1]

Expectation of integrable random variables is unchanged by almost-sure replacement (Expectation depends only on the almost-everywhere class).

[L2]

A nonnegative measurable function has integral zero exactly when it is zero almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[L3]

Convergence in probability means that every fixed positive tail probability tends to zero (Convergence in probability).

Proof

technique · direct

1.1

The integrand is bounded by $1$ , so its expectation is finite. Almost-sure replacement of either representative leaves it unchanged almost surely, hence leaves its expectation unchanged by [L1]. Symmetry is immediate; and [L1] $\min (1, ∣ X - Z ∣) \leq \min (1, ∣ X - Y ∣) + \min (1, ∣ Y - Z ∣)$ by the real triangle inequality. Taking expectations gives the triangle inequality.

1.2

If $d_{0} ([X], [Y]) = 0$ , [L2] makes $\min (1, ∣ X - Y ∣) = 0$ almost surely. [L2] $X = Y$ almost surely; the converse is clear. Thus $d_{0}$ is a metric.

1.3

For $0 < ε \leq 1$ , splitting at the error event gives. [algebra] $ε P (∣ X_{n} - X ∣ > ε) \leq d_{0} ([X_{n}], [X])$ and $d_{0} ([X_{n}], [X]) \leq ε + P (∣ X_{n} - X ∣ > ε) .$ The first comes from the bad set; the second splits it from its complement.

algebra

2.1

The first inequality makes $d_{0} \to 0$ imply probability convergence by [L3]. Conversely, [L3] and the second inequality give $lim sup d_{0} \leq ε$ for every $ε > 0$ , hence $d_{0} \to 0$ .

step 1.3L3∎

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Cited to discharge well-definedness by A metric for convergence in probability.

Dependency tree · two levels

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Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Exercise 3.2.8 (comparison metric) (standard reference, not scraped)

Convergence in probability is metrized by d0d0​