Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution.