I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\mathbb R$-valued random variables independent of the $X_i$s with finite second moments (w.l.o.g., assume $\mathbb EW_i = 0$ and $\mathbb EW_i^2 = 1$). Consider the sequence $(Z_n)_{n\in\mathbb N}$ of $\mathbb R$-valued random variables given by $$Z_n := \frac 1{\sqrt n}\sum^n_{i=1}W_iX_i.$$ Intuitively, this sequence should converge, conditional on the $X_i$s, to $\mathcal N(0,\sigma^2)$, where $\sigma^2 := \mathbb EX_i^2 - (\mathbb EX_i)^2$. However, I don't know how to show this rigorously. Part of the problem is that I don't even have a definition...

I have seen this definition: $(Z_n)_{n\in\mathbb N}$ converges weakly conditional on the $X_i$s to a $\mathbb R$-valued random variable $Z$ iff $(\Delta_n)_{n\in\mathbb N}$ converges to zero in probability for all bounded, continuous $\mathbb R$-valued functions on $\mathbb R$, where $$\Delta_n := \left\vert\mathbb E[f(Z_n)\mid\mathcal H] - \mathbb E[f(Z)]\right\vert$$ with $\mathcal H := \sigma(\{X_i : i\in\mathbb N\})$ denoting the $\sigma$-algebra generated by the $X_i$s. Are there other (equivalent but more straight forward) definitions of weak conditional convergence?

However, this definition does not really help me as computing the conditional expectation of $f(Z_n)$ given $\mathcal H$ for every bounded, continuous $\mathbb R$-valued functions is exhaustive.

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