Large Deviation Tail Estimates and Related
Limit Laws for Stochastic Fixed Point Equations
Jeffrey F. Collamore and Anand N. Vidyashankar
Abstract We study the forward and backward recursions generated by a stochastic
fixed point equation (SFPE) of the form V
d
D A maxfV;DgCB , where .A; B; D/ 2
.0; 1/ R
2
, for both the stationary and explosive cases. In the stationary case
(when EŒlog A < 0/, we present results concerning the precise tail asymptotics
for the random variable V satisfying this SFPE. In the explosive case (when
EŒlog A > 0/, we establish a central limit theorem for the forward recursion
generated by the SFPE, namely the process V
n
D A
n
maxfV
n1
;D
n
gC B
n
, where
f.A
n
;B
n
;D
n
/ W n 2 Z
C
g is an i.i.d. sequence of random variables. Next, we
consider recursions where the driving sequence of vectors, f.A
n
;B
n
;D
n
/ W n 2
Z
C
g, is modulated by a Markov chain in general state space. We demonstrate an
asymmetry between the forward and backward recursions and develop techniques
for estimating the exceedance probability. In the process, we establish an interesting
connection between the regularity properties of fV
n
g and the recurrence properties
of an associated -shifted Markov chain. We illustrate these ideas with several
examples.
J.F. Collamore ()
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5,
DK-2100 Copenhagen Ø, Denmark
e-mail: collamore@math.ku.dk
Research supported in part by Danish Research Council (SNF) Grant No. 09-092331.
A.N. Vidyashankar
Department of Statistics, Volgeneau School of Engineering, George Mason University,
Fairfax, VA 22030, USA
e-mail: avidyash@gmu.edu
Research supported in part by NSF grant DMS 1107108.
G. Alsmeyer and M. L¨ owe (eds.), Random Matrices and Iterated Random Functions,
Springer Proceedings in Mathematics & Statistics 53,
DOI 10.1007/978-3-642-38806-4 5, © Springer-Verlag Berlin Heidelberg 2013
91