### Concours Mines-Ponts: mathématiques 2 PC

The subject was Markov chains. A discrete system was modeled, with the number of transitions in the interval \([0,t]\) following a Poisson distribution. Quite a few tricky sum calculations, but the subject seemed a little simpler than the 1st, and in any case more classical.

Positive symmetrical matrices were used again, in the form of positive self-adjoint endomorphisms, which was a little redundant with Topic 1.

Question 18 was the fundamental result of the last part, and allowed us to determine in question 21 that the probability law of the system stabilized after a sufficiently long time: \[\forall j\in \lBrack 1,N\rBrack,\quad \lim_{t\to+\infty} H_t[1,j] = \pi(j) \], where

- \(H_t[1,j]\) is the probability that the system is in state \(j\) at time \(t\)
- \(\pi\) is the invariant probability of the transition matrix \(K\) (eigenvector on the left associated with eigenvalue 1).

A complete set of answers for the Maths 2 test on May 3 in the PC concours Mines-Ponts 2023:

Find the problem set here: