  Normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen
(1997) by subordinating Brownian motion with drift to an inverse Gaussian
process. Increments of NIG process are independent and stationary. In this
paper, we introduce dependence between the increments of NIG process, by
subordinating fractional Brownian motion to an inverse Gaussian process and
call it fractional normal inverse Gaussian (FNIG) process. The basic properties
of this process are discussed. Its marginal distributions are scale mixtures of
normal laws, infinitely divisible for the Hurst parameter 1/2<=H< 1 and are
heavy tailed. First order increments of the process are stationary and possess
long-range dependence (LRD) property. It is shown that they have persistence of
signs LRD property also. A generalization of the FNIG process called n-FNIG
process is also discussed which allows Hurst parameter H in the interval (n-1,
n). Possible applications to mathematical finance and hydraulics are also
pointed out
