A Survey on Conditioned Limit Theorems for Products of Random Matrices and Affine Random Walks

Limit theorems for affine Markov walks conditioned to stay positive

“…We refer the reader to Iglehart [18], Bolthausen [2], Doney [11], Bertoin and Doney [1], Borovkov [4,3], Caravenna [5], Eichelsbacher and Köning [12], Garbit [13], Denisov, Vatutin and Wachtel [7], Denisov and Wachtel [8,10]. More general walks with increments forming a Markov chain have been considered by Presman [20,21], Varapoulos [22,23], Dembo [6], Denisov and Wachtel [9] or Grama, Le Page and Peigné [15]. In [20,21] the case of sums of lattice random variables defined on finite regular Markov chains has been considered.…”

“…Varapoulos [22,23] studied Markov chains with bounded increments and obtained lower and upper bounds for the probabilities of the exit time from cones. Some studies take advantage of additional properties: for instance in [9] the Markov walk has a special integrated structure; in [15] the moments of X n are bounded by some constants not depending on the initial condition. However, to the best of our knowledge, the asymptotic behaviour of the probability P x (τ y > n) in the case of the stochastic recursion (1.1) has not yet been considered in the literature.…”

“…Note that the Wiener-Hopf factorization, which usually is employed in the case of independent random variables, cannot be applied in a straightforward manner for Markov chains. Instead, to study the case of the stochastic recursion, we rely upon the developments in [9], [10] and [15]. The main idea of the paper is given below.…”

“…While the harmonicity is inherently related to the martingale properties, the difficulty is to show that the approximating martingale is integrable at the exit time of the Markov walk (y + S n ) n 1 . In contrast to [10] and [15], our proof of the existence of V employs different techniques according to positivity or not of the values of E(a 1 ). The constructed harmonic function allows to deduce the properties of the exit time and the conditional distribution of the Markov walk from those of the Brownian motion using a strong approximation result for Markov chains from Grama, Le Page and Peigné [16].…”

“…Then, by the Markov property, we are able to reduce the problem to a Markov walk with a large starting point y ′ = y + S νn , where ν n is the first time when the sequence |y + S k | exceeds the level n 1/2−ε . The major difficulty, compared to [10] and [15], is that, for the affine model under consideration, the sequence (X νn ) n 1 is not bounded in L 1 . To overcome this we need a control of the moments of X n in function of the initial state X 0 = x and the lag n.…”