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A chi-square variable divided by a comparable dimension converges to one

Statement

Assume AC. Let $r_{k}, n_{k}$ be positive integers with $n_{k} \to \infty$ and $r_{k} / n_{k} \to 1$ . For each $k$ , let $Z_{k, 1}, \dots, Z_{k, r_{k}}$ be independent $N (0, 1)$ random variables and set $D_{k} = \sum_{j = 1}^{r_{k}} Z_{k, j}^{2}$ . Then $D_{k} / n_{k} \to 1$ in probability.

Statement references

Wainwright, Appendix K, chi-square concentration

Facts & Assumptions

Given: AC, integer sequences $r_{k}, n_{k}$ with $n_{k} \to \infty$ and $r_{k} / n_{k} \to 1$ , and independent standard-normal variables $Z_{k, j}$ .

[F1]

A standard normal has density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ (Standard normal and normal laws).

[F2]

For a nonnegative random variable $U$ and $a > 0$ , $P (U \geq a) \leq E [U] / a$ (Markov's inequality for random variables).

[A1]

AC supplies the finite independent normal coordinates used in the statement (The Axiom of Choice).

Proof

1.1A1F1algebra

Integration by parts against [F1], using $ϕ^{'} (x) = - x ϕ (x)$ and the vanishing Gaussian boundary terms, gives $E Z_{k, j}^{2} = 1$ and $E Z_{k, j}^{4} = 3$ , hence $Var (Z_{k, j}^{2}) = 2$ .

2.1F2step 1.1givenalgebra∎

Independence and step 1.1 give $E D_{k} = r_{k}$ and $E (D_{k} - r_{k})^{2} = 2 r_{k}$ . Fix $η > 0$ ; eventually $∣ r_{k} / n_{k} - 1 ∣ < η / 2$ , so $∣ D_{k} / n_{k} - 1 ∣ > η$ implies $∣ D_{k} - r_{k} ∣ > η n_{k} / 2$ . Applying [F2] to $(D_{k} - r_{k})^{2}$ gives $P (∣ D_{k} / n_{k} - 1 ∣ > η) \leq 8 r_{k} / (η^{2} n_{k}^{2}) \to 0$ , since $r_{k} / n_{k} \to 1$ and $n_{k} \to \infty$ . This is convergence in probability.

Source notes

The cited appendix supplies the Gaussian matrix concentration context. The displayed ratio conclusion is proved here directly from the standard-normal second and fourth moments.

Depends on

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Sources

Wainwright, Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using L1-Constrained Quadratic Programming (Lasso), 2009, Appendix K (standard reference, not scraped)