# Limit distribution of gaussian norm and Paley-Zygmund inequality (solution verification)

Let $$X_i \sim \mathcal N(0, I_{d(n)})$$ be iid normal random vectors taking values in $$\mathbb R^{d(n)}$$ where $${d(n)}$$ is a sequence (dependent on a positive integer $$n$$) that grows such that $${d(n)}/n \rightarrow \gamma >0$$ as $$n\rightarrow \infty$$ for some positive constant $$\gamma \in (0,1)$$. Consider the following quantity:

$$Y_n = \left\|\frac{1}{\sqrt n} \sum_i^n X_i\right \|_2^2$$

Does $$Y_n$$ have a limiting distribution? If so, what is it?

What I've tried

Rewriting, we have:

$$Y_n = \left\|\sqrt n\,\frac{1}{n} \sum_i^n X_i\right \|_2^2 = \left\|\sqrt n\, \bar X\right \|_2^2$$

where $$\bar X \sim \mathcal N\left(0, I_{d(n)}/n\right)$$ so that $$Y_n$$ is clearly chi squared distributed with $$d(n)$$ degrees of freedom. The chi squared distribution does not have a limiting distribution as the number of degrees of freedom increases to $$\infty$$. This shows that $$Y_n$$ does not converge in distribution.

The homework problem suggests to apply the Paley-Zygmund inequality, but I don't understand where that might be relevant. Any hints would be appreciated.

Paley-Zygmund:

$$\mathbb P[X\ge\theta \mathbb EX]\ge (1-\theta)^2\frac{\mathbb E[X]^2}{\mathbb E[X^2]}$$

• Hint: Have you computed the mean and variance of $Y_n$ and then plug into the bound?
– E-A
Mar 9, 2021 at 21:22
• @E-A $P(Y_n \geq \theta d) \geq d^2/(2d+d^2) \rightarrow 0$ but I don't see how this lower bound tells me anything about the convergence in distribution.
– dmh
Mar 9, 2021 at 21:29
• That limit -assuming your expression is right- is 1/2, not 1? The point is that you can show that this violates tightness, i.e. you have some part of your measure escaping to infinity.
– E-A
Mar 9, 2021 at 22:32
• Oh that's great, I messed up that limit calculation. I now see how it violates tightness.
– dmh
Mar 9, 2021 at 23:00

$$\mathbb P[Y_n\ge\theta d]\ge (1-\theta)^2\frac{d^2}{2d + d^2} \rightarrow 1/2$$
and so the limit distribution of $$Y_n$$ is not tight and so does not exist.