## Limit theorems for quadratic forms with applications to Whittle’s estimate.(English)Zbl 0940.60037

This paper corrects the limit distribution of a quadratic form statistic previously announced by J. Beran and N. Terrin [Biometrika 83, No. 3, 627-638 (1996; Zbl 0866.62055)]. This difference of quadratic forms can be used in determining the presence of a change-point in the long-memory parameter of a dependent sequence of random variables. The underlying data, $$X_i$$, are assumed to be a moving average of iid variables. Define $Q^*(x,y) =\sum_{[x]<i,j\leq [y]}b(i-j)\{X_iX_j-r(i-j)\}$ for a suitable sequence of constants $$b(k)$$ and where $$r(k)$$ denotes the covariance of $$X_0$$ and $$X_k$$. Define $$Q(x)=Q^*(x,\infty)$$ and $Z_n(t)=n^{-1/2}t(1-t)\{Q(nt)/nt - Q^*(nt,n)/n(1-t)\}.$ Under decay conditions on the moving average weights and the constants $$b$$, the paper establishes a strong approximation of $$Q(x)$$ by a Brownian motion, and of $$Z_n$$ by a sequence of independent Brownian bridges. It then obtains a strong law of large numbers for $$Q(n)$$. The paper examines the case where the moving average coefficients for $$X_k$$ depend on a finite-dimensional parameter, $$\lambda_k$$. These are assumed to be constant up to a certain point, and then equal to a different constant after that change-point. Splitting the data at $$k$$, the Whittle estimate of $$\lambda$$ before $$k$$ is $$\widehat{\lambda}_k$$: that after, $$\widetilde{\lambda}_k$$. Theorem 2.1 obtains that under the null hypothesis that there is no change-point, the normalized difference $$\widehat{\lambda}_{nt}-\widetilde{\lambda}_{nt}$$ converges weakly to a scaled Brownian motion. Theorem 2.2 provides a similar result for weighted metrics. Theorem 2.3 provides upper tail critical points. The proofs decompose the quadratic forms into martingales and smaller terms.

### MSC:

 60F17 Functional limit theorems; invariance principles 60G70 Extreme value theory; extremal stochastic processes 60F05 Central limit and other weak theorems 60F15 Strong limit theorems