On recurrence and transience of multivariate near-critical stochastic processes

Abstract: We obtain complementary recurrence and transience criteria for processes

X = (X_{n})_{n g e 0}

with values in

m a t h b b R_{+}^{d}

fulfilling a non-linear equation

X_{n + 1} = M X_{n} + g (X_{n}) + x i_{n + 1}

. Here

M

denotes a primitive matrix having Perron-Frobenius eigenvalue 1, and

g

denotes some function. The conditional expectation and variance of the noise

(x i_{n + 1})_{n g e 0}

are such that

X

obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [5].