◦ *Markov chain with two states of order 1*

For a first order Markov chain, the state of the variable *E*(*t*) at time *t* depends only on its state at time (*t* − 1). Thus, we have four situations: [31]

$$\begin{aligned} \text{P}\_{00} &= \text{pr}(\text{E}(\text{t}+1) = 0 \mid \text{E}(\text{t}) = 0)) \\ \text{P}\_{01} &= \text{pr}(\text{E}(\text{t}+1) = 1 \mid \text{E}(\text{t}) = 0)) \\ \text{P}\_{10} &= \text{pr}(\text{E}(\text{t}+1) = 0 \mid \text{E}(\text{t}) = 1)) \\ \text{P}\_{11} &= \text{pr}(\text{E}(\text{t}+1) = 1 \mid \text{E}(\text{t}) = 1)) \end{aligned} \tag{4}$$

Pij is the probability of going to state *j* knowing that you are in state *i*. These probabilities were calculated using the following relationship:

$$P\_{\vec{\text{ij}}} = \mathbb{N}\_{\vec{\text{ij}}} / \mathbb{N}\_{\vec{\text{i}}} \text{ with: } \text{i and } \mathbf{j} = 0 \text{ or } 1 \tag{5}$$

Nij is the transition number from state *i* to state *j* and Ni is the number of transitions from state *i* to any other state. The pairs of years Nij are determined [35] (Equation (6)):

$$\begin{cases} \mathbf{N}\_0 = \mathbf{N}\_{00} + \mathbf{N}\_{01} \\ \mathbf{N}\_1 = \mathbf{N}\_{10} + \mathbf{N}\_{11} \\ \mathbf{N} = \mathbf{N}\_0 + \mathbf{N}\_1 \end{cases} \tag{6}$$

N0; N1 and N are the number of dry, wet years and the total number of years of observation, respectively. N01 and N10 respectively represent the number of years of state change from a dry year to a wet year and from a wet year to a dry year. The transition matrix P of the conditional probabilities Pij, is presented so that each line is equal to 1 [35]. Resulting in a set of possible Pij values (Equation (7)):

$$P = \begin{bmatrix} P\_{00} & P\_{01} & \dots \\ P\_{10} & P\_{11} & \dots \\ \dots & \dots & \dots \\ P\_{i0} & P\_{i1} & \dots \end{bmatrix} \tag{7}$$
