TY - JOUR
T1 - On the Growth Rate of a Linear Stochastic Recursion with Markovian Dependence
AU - Pirjol, Dan
AU - Zhu, Lingjiong
N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.
PY - 2015/9/1
Y1 - 2015/9/1
N2 - We consider the linear stochastic recursion $$x_{i+1} = a_{i}x_{i}+b_{i}$$xi+1=aixi+bi where the multipliers $$a_i$$ai are random and have Markovian dependence given by the exponential of a standard Brownian motion and $$b_{i}$$bi are i.i.d. positive random noise independent of $$a_{i}$$ai. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $$\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]$$λq=limn→∞1nlogE[(xn)q] with $$q\in \mathbb {Z}_+$$q∈Z+. We show that the Lyapunov exponents $$\lambda _q$$λq exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.
AB - We consider the linear stochastic recursion $$x_{i+1} = a_{i}x_{i}+b_{i}$$xi+1=aixi+bi where the multipliers $$a_i$$ai are random and have Markovian dependence given by the exponential of a standard Brownian motion and $$b_{i}$$bi are i.i.d. positive random noise independent of $$a_{i}$$ai. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $$\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]$$λq=limn→∞1nlogE[(xn)q] with $$q\in \mathbb {Z}_+$$q∈Z+. We show that the Lyapunov exponents $$\lambda _q$$λq exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.
KW - Critical exponent
KW - Large deviations
KW - Linear stochastic recursion
KW - Lyapunov exponent
KW - Phase transitions
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U2 - 10.1007/s10955-015-1280-3
DO - 10.1007/s10955-015-1280-3
M3 - Article
AN - SCOPUS:84938421636
SN - 0022-4715
VL - 160
SP - 1354
EP - 1388
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 5
ER -