Invariance principle for the least squares estimates (Q1332027)

The regression \(x(i,j)= \theta\varphi (i,j)+ \varepsilon(i,j)\) is considered where \(\varphi(i,j): N^ 2\to R^ 1\) is a known function, \(\varepsilon(i,j)\) is a strong martingale difference, and \(\theta\) is an unknown parameter. The least squares estimator of the parameter \(\theta\) has the form \[ \widehat {\theta} (n,m)= d^{-2} (n,m) \sum_{i=1}^ n \sum_{j=1}^ m \varphi(i,j) x(i,j), \qquad \text{where} \qquad d^ 2(n,m)= \sum_{i=1}^ n \sum_{j=1}^ m \varphi^ 2 (i,j). \] Let \[ \begin{aligned} X_ n(t,s) &= d^ 2 ([nt], [ns]) d^{-1} (n,n) (\widehat {\theta} ([nt], [ns])- \theta),\\ X_ n(0,s) &\equiv X_ n (t,0)\equiv X_ n (0,0) =0; \qquad t,s\in [0,1].\end{aligned} \] Under these conditions and if \[ \varliminf_{n\to\infty} d^ 2 ([nt], [ns]) d^{-2} (n,n)= \sigma^ 2 (t,s)\in \mathbb{C} ([0,1]^ 2), \] then the sequence of distributions of the random fields \(X_ n (t,s)\) converges weakly to the distribution of the random field \(X(t,s)\) with independent Gaussian increments and with \(E X(t,s)=0\), \(E X^ 2 (t,s)= \sigma^ 2 (t,s)\) in \(D([0,1]^ 2)\).