TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-07

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Slutsky's theorem for real random variables

Statement

Let $(X_{n})$ and $(Y_{n})$ be real random variables on one probability space, and let $X$ be a real random variable (possibly on another space). If $X_{n} \Rightarrow X$ and $Y_{n} \to c$ in probability for $c \in R$ , then $X_{n} + Y_{n} \Rightarrow X + c$ and $X_{n} Y_{n} \Rightarrow c X$ . If $c \neq 0$ , define $Q_{n} = X_{n} / Y_{n}$ on ${Y_{n} \neq 0}$ and give $Q_{n}$ any fixed value on ${Y_{n} = 0}$ . Then $Q_{n} \Rightarrow X / c$ .

Facts & Assumptions

Given: $(X_{n})$ and $(Y_{n})$ are on one probability space; $X_{n} \Rightarrow X$ , $Y_{n} \to c$ in probability, and the displayed quotient convention when $c \neq 0$ .

[L1]

Distributional convergence is CDF convergence at continuity points (Convergence in distribution for real random variables).

[L2]

Probability convergence makes $P (∣ Y_{n} - c ∣ > δ) \to 0$ for each $δ > 0$ (Convergence in probability).

Proof

technique · direct

1.1L1L2

For real $U_{n}, V_{n}$ on a common probability space, if $U_{n} \Rightarrow U$ and $∣ V_{n} - U_{n} ∣ \to 0$ in probability, then $V_{n} \Rightarrow U$ . Indeed, for every $δ > 0$ , with $p_{n} = P (∣ V_{n} - U_{n} ∣ > δ)$ , $F_{U_{n}} (t - δ) - p_{n} \leq F_{V_{n}} (t) \leq F_{U_{n}} (t + δ) + p_{n} .$ At a continuity point $t$ of $F_{U}$ , take $δ ↓ 0$ through values for which both $t - δ$ and $t + δ$ are continuity points. These values exist because a CDF has at most countably many jumps (for each positive integer $k$ , there are at most $k$ jumps larger than $1 / k$ ). First let $n \to \infty$ for each such $δ$ , then let $δ ↓ 0$ ; [L1] and [L2] give the assertion.

1.2L1

The CDF definition [L1] gives both affine operations needed below. First, $X_{n} + a \Rightarrow X + a$ because $F_{X_{n} + a} (t) = F_{X_{n}} (t - a)$ . It also gives $a X_{n} \Rightarrow a X$ for every constant $a$ : for $a > 0$ use $F_{a X_{n}} (t) = F_{X_{n}} (t / a)$ ; for $a < 0$ , use $F_{a X_{n}} (t) = 1 - F_{X_{n}} ((t / a) -)$ and squeeze the left limit between $F_{X_{n}} (t / a - δ)$ and $F_{X_{n}} (t / a)$ , taking $δ ↓ 0$ through continuity points $t / a - δ$ ; and for $a = 0$ the claim is immediate. At continuity points of the transformed limit CDF, the corresponding point of $F_{X}$ is a continuity point.

1.3L1L2

The sequence $(X_{n})$ is bounded in probability: CDF convergence [L1] at two continuity points outside a sufficiently large interval makes ${lim sup}_{n} P (∣ X_{n} ∣ > M)$ arbitrarily small. Therefore $P (∣ X_{n} (Y_{n} - c) ∣ > ε) \leq P (∣ X_{n} ∣ > M) + P (∣ Y_{n} - c ∣ > ε / M)$ shows $X_{n} (Y_{n} - c) \to 0$ in probability. If $c \neq 0$ , on $∣ Y_{n} - c ∣ < ∣ c ∣ / 2$ , $∣ Q_{n} - X_{n} / c ∣ \leq \frac{2}{∣ c ∣^{2}} ∣ X_{n} ∣ ∣ Y_{n} - c ∣,$ and the exceptional event ${∣ Y_{n} - c ∣ \geq ∣ c ∣ / 2}$ contains ${Y_{n} = 0}$ and has probability at most $P (∣ Y_{n} - c ∣ > ∣ c ∣ / 4) \to 0$ ; the same boundedness argument gives $Q_{n} - X_{n} / c \to 0$ in probability.

2.1step 1.1step 1.2L2

Addition follows from step 1.1 with $U_{n} = X_{n} + c$ and $V_{n} = X_{n} + Y_{n}$ : $U_{n} \Rightarrow X + c$ by step 1.2, while $∣ V_{n} - U_{n} ∣ = ∣ Y_{n} - c ∣ \to 0$ in probability by [L2].

3.1step 1.1step 1.2step 1.3∎

Apply step 1.1 to $U_{n} = c X_{n}$ , $V_{n} = X_{n} Y_{n}$ , using step 1.2 and step 1.3, to obtain $X_{n} Y_{n} \Rightarrow c X$ . When $c \neq 0$ , apply it again to $U_{n} = X_{n} / c$ , $V_{n} = Q_{n}$ , to obtain $Q_{n} \Rightarrow X / c$ .

Depends on

Used by

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Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 3.2, Exercises 3.2.12--3.2.14 (standard reference, not scraped)