Let Z(t)=(Z1(t),…,Zd(t))⊤,t∈R where Zi(t),t∈R, i=1,…,d are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For X(t)=AZ(t),t∈R, A is a nonsingular d×d real-valued matrix, u,c∈Rd T>0 we derive tight bounds for P∃t∈[0,T]:∩i=1d{Xi(t)−cit>ui}and find exact asymptotics as (u1,…,ud)⊤=(ua1,…,uad)⊤ any (a1,…,ad)⊤∈Rd∖(−∞,0]d u→∞.

In this article, normal inverse Gaussian (NIG) autoregressive model is introduced. The parameters of the are estimated using expectation maximization (EM) algorithm. efficacy EM algorithm shown simulated and real-world financial data. It that NIG fit very well on considered data hence could be useful in modelling various real-life time-series