On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations

Let xₜ (t = 1, 2, ) be defined recursively by equation*1. 1xₜ = axₓ-₁ + uₜ, t = 1, 2, , equation* where x₀ is a constant, uₜ = 0, u²ₜ = ² and uₜuₛ = 0, t s. (denotes mathematical expectation. ) An estimate of based on x₁, , xT (which is the maximum likelihood estimate of if the u's are normally distributed) is equation*1. 2 = (Tₓ=₁ xₜxₓ-₁) / (Tₓ=₁ x²ₓ-₁). equation* If || 0. (See 2, Chapter II, for example. ) If || > 1, White 3 has shown (-) ||T/ (² - 1) has a limiting Cauchy distribution under the assumption that x₀ = 0 and the u's are normally distributed; he has also found the distribution when x₀ 0. His results can be easily modified and restated in the following form (Tₓ=₁ x²ₓ-₁) ^1{2} (-) has a limiting normal distribution if the u's are normally distributed and if || 1. Peculiarly, for || = 1 this statistic has a limiting distribution which is not normal (and is not even symmetric for x₀ = 0). One purpose of this paper is to characterize the limiting distributions for || > 1 when the u's are not necessarily normally distributed; it will be shown that for || > 1 the results depend on the distribution of the u's. Central limit theorems are not applicable. Secondly, the limiting distribution for || < 1 will be shown to hold under the assumption that the u's are independently, identically distributed with finite variance. This was conjectured by White.