= Solution
A useful <method of moments in probability> states that if real <random variables> $Z_N$ have all moments, $\mathbb E Z_N^j\to m_j$ for every integer $j\ge1$, and $(m_j)$ is the moment sequence of a <moment-determinate probability distribution>, then $Z_N$ converges to that distribution in the sense of <convergence in distribution>. In particular, convergence to the <standard normal distribution> follows from limiting odd moments zero and limiting even moments $(2k-1)!!$. The <moment-generating function of a standard normal variable>, finite in a neighbourhood of zero, ensures <moment determinacy>.
For the classical <Erdős-Kac theorem>, put $L=\log\log N$ and choose, for example, $\varphi(N)=L^{1/4}$, with a harmless modification for small $N$. Truncate the <prime omega function> to <primes> $p\le y=N^{1/\varphi(N)}$. The omitted number of distinct <prime factors> of any $n\le N$ is at most $\varphi(N)$, since a product of $r$ omitted <primes> exceeds $N^{r/\varphi(N)}$. Thus this truncation changes the normalized variable by $o(1)$ uniformly. Also, the <Mertens theorem for reciprocal primes> gives
$$
\sum_{p\le y}\frac1p=L-\log\varphi(N)+O(1)=L+o(\sqrt L).
$$
Part (b) transfers every fixed normalized <central moment> of the truncated <prime omega function> to the independent <Bernoulli random variables> model. Indeed, its unnormalized error is at most $O_j(N^{2j/\varphi(N)-1})$, using $\sum_{p\le y}1\le y$, and this tends to zero for each fixed $j$. In the independent expansion, any singleton index has zero <expectation>; the leading contributions are pairings, giving exactly the <standard normal distribution> moments, while blocks of size at least three are negligible after normalization. The <method of moments in probability>, followed by the uniformly negligible truncation and centering errors, proves the <Erdős-Kac theorem>.
The same strategy proves the bounded-prime-weight form in part (a). Choose $\varphi(N)\to\infty$ slowly enough that $\varphi(N)=o(B(N))$. The omitted weighted contribution is $O(\varphi(N))$; its mean is likewise $O(\varphi(N))$, and its independent <variance> is $O(\log\varphi(N)+1)=o(B(N)^2)$ by the <Mertens theorem for reciprocal primes>. Hence the truncated model has the same normalization, and part (b) again transfers each fixed <central moment>.
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