# CLT for inner product of Hilbert space valued random variables

Let $$\mathcal{H}$$ be a Hilbert space over $$\mathbb{R}$$ with inner product $$\langle \cdot , \cdot \rangle$$ and let $$(X_n)_{n \in \mathbb{N}}$$ and $$(Y_n)_{n \in \mathbb{N}}$$ be i.i.d sequences of random variables with values in $$\mathcal{H}$$. Assume that $$E\langle X_n, Y_n \rangle = 0$$, $$0 < E( \lVert X_n \rVert^2 \lVert Y_n \rVert^2)< \infty$$ and that the sequence of inner products is independent. Does

$$\frac{1}{\sqrt{n}} \sum_{i=1}^n \langle X_i, Y_i \rangle \overset{\mathcal{D}}{\to} N(0, E((\langle X_1, Y_1 \rangle)^2))$$

then follow from the classical CLT?

Let $$Z_n = \langle X_n, Y_n\rangle$$. We know that the sequence $$(Z_n)$$ is independent, and that $$E Z_n = 0$$ and $$\text{Var}(Z_n) \le E(|X_n|^2 |Y_n|^2) < \infty$$. If the $$Z_n$$ also are identically distributed (which you seem to assume), then your result is simply an application of the classical CLT on the sequence $$(Z_n)$$.
• I don't see where OP states that $Z_n$'s are identically distributed. Classical CLT does not apply. – Kavi Rama Murthy Feb 11 at 23:19