# Through-the-cycle rating transition matrix

Suppose we know the observed transition matrix for annual migrations between credit ratings, $$T_{ij,t}$$, for $$N$$ years. How is the through-the-cycle (TTC) transition matrix defined?

Sometimes the following definition is used, just like other TTC quantities which are long-run averages (through a full economic cycle): $$\overline{T}_{ij} = \mathbb{E}(T_{ij,t})$$

However, since the dynamics of credit rating migrations comes about by multiplying transition matrices, the above definition in terms of an arithmetic mean looks problematic in my opinion.

Shouldn't a more accurate definition be given in terms of the geometric mean? $$\overline{T} = \left(\prod_{t=1}^{N}(T_{t})\right)^{1/N}$$ or, alternatively $$\overline{\log T} = \mathbb{E}(\log T_{t})$$ Now, it is well known that taking logs and roots of transition matrices is fraught with non-uniqueness and non-positivity issues but there are some ways to tackle this -- there is some discussion here.

My question is what should be the correct definition of the through-the-cycle transition matrix? (For those who can relate, this question is similar to calculating quenched vs. annealed averages for disordered systems in physics.)