Interpolation of random functions (Q1183046)

Let \((\Omega,A,P)\) be a complete probability space and \(X=X(t,\omega)\), \(t\in[a,b]\subseteq R\) and \(\omega\in\Omega\), a family of stochastic processes. The aim of this paper is to obtain error bounds for the \(p\)th absolute moment of the quantity \(\sup_{t\in[a,b]}| X(t)- (TX)(t)|\), for \(T\) an interpolation operator. The case of interpolation by cubic spline is also considered. If the step sizes \(h_ n\) verify \(\sum^ \infty_{n=1}h^ 7_ n<\infty\) then \(\lim_{n\to\infty}\sup_{t\in[a,b]}| X(t)-(T_{h_ n}X)(t)|=0\), with probability 1, for every separable process \(X\) having the derivatives \(X^{(k)}\) in \(C([a,b],\;L^ 2(\Omega,A,P))\), for \(0\leq k\leq 4\).