Sampling distribution for a class of estimators for nonregular linear processes

Let {Xt; T = 1, 2,...} be a linear process with a location parameter [theta] defined by Xt - [theta] = [Sigma]0[infinity]grZt-r where {Zt; T = 0, ±1,...} is a sequence of independent and identically distributed random variables, with E[short parallel]Z1[short parallel][delta] < [infinity] for some [delta] > 0. If [delta] [greater-or-equal, slanted] 1 we assume further than E(Z1) = 0. Let [eta] = [delta] if 0 < [delta] < 2, and [eta] = 2 if [delta] [greater-or-equal, slanted] 2. Then assume that [Sigma]0[infinity][short parallel] gr [short parallel][eta] < [infinity]. Consider the class of estimators given by is of the form cnt = [Sigma]p = 0s[beta]nptp for some s [greater-or-equal, slanted] 0. An attempt has been made to investigate the distributional properties of in large samples for various choices of [beta]np (0 [less-than-or-equals, slant] p [less-than-or-equals, slant] s), s, and the distribution of Z1 under the constraints [Sigma]0[infinity]rkgr = 0, 0 [less-than-or-equals, slant] k [less-than-or-equals, slant] q where q in an arbitrary integer, 0 [less-than-or-equals, slant] q [less-than-or-equals, slant] s.