  We develop the relevant combinatorics pertaining to cyclic-conditional
freeness in order to introduce the adequate sets of cumulants linearizing the
cyclic-conditional additive convolution. On our way, we introduce a new
non-commutative independence, cyclic freeness. We explain how
cyclic-conditional freeness is "reduced" to cyclic freeness by utilizing a
multivariate extension of the inverse Markov-Krein transform. We also consider
cyclic-conditional multiplicative convolution and prove limit theorems.
