Moment generating function and generalized Feller operators (Q2721804)

Let \(\{X_i\}\) be the sequence of random variables with independent identical distribution (denoted by i.i.d.r.v sequence), \(EX_i=x\), \(x\in(-\infty, +\infty)\), we denote by \(S_n=X_1+ X_2+\cdots +X_n\), the distribution function of \(S_n\) be \(F_{n,x}(t)\), for \(j\in C_B(-\infty, +\infty)\) Feller operators are defined by NEWLINE\[NEWLINEF_n(f_ix): =Ef\left({S_n\over n}\right)= \int^\infty_{-\infty} f\left({t\over n} \right)dF_{n,x}(t),NEWLINE\]NEWLINE where \(E\) denotes mathematical expectation. If mathematical expectation \(EX_i\varphi(x)\), the authors generalize the Feller operators: let \(\{X_i\}\) be i.i.d.r.v sequence, \(EX_i= \varphi(x)\), \(\varphi(x\) be strictly increasing function on \([0,R)\) \((R\) is a finite positive number or \(+\infty)\), its inverse function \(\psi(x)\), \(\text{Var} X_i= \sigma^2(x)< +\infty\), generalized Feller operators are defined by NEWLINE\[NEWLINEL_n(f;x): =Ef\left[\psi \left({S_n\over n}\right) \right]= \int_0^{+ \infty} f\left[\psi \left({S_n\over n}\right)\right]d F_{n,x}(t),\;x\in [0,R),NEWLINE\]NEWLINE the special case of \(L_n(f;x)\) is the Meyer-König and Zeller operator. Applying the calculative property of moment generating functions and probabilistic methods, in the paper the convergence and estimation of approximation degree to unbounded functions for generalized Feller operators are obtained.