On the Order of Approximation in Limit Theorems for Negative–Binomial Sums of Strictly Stationary m-Dependent Random Variables

• Published in: Acta mathematica vietnamica Vol. 46; no. 1; pp. 203 - 224
• Main Authors: Hung, Tran Loc, Kien, Phan Tri
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.03.2021; Springer Nature B.V
• Subjects: Approximation; Characteristic functions; Dependent variables; Independent variables; Mathematical analysis; Mathematics; Mathematics and Statistics; Probability theory; Queuing theory; Random variables; Sums; Theorems; Geometric sums; 41A25; Primary 60E07; Negative–binomial sums; Dependent random variables; Moving average processes; Strictly stationary sequence; Zolotarev’s probability metric; 60F05; Secondary 62E17
• ISSN: 0251-4184; 2315-4144
• DOI: 10.1007/s40306-020-00406-x

During the last several decades, the results related to geometric random sums of independent identically distributed random variables have become interesting results in probability theory and in insurance, risk theory, queuing theory and stochastic finance, etc. The negative–binomial random sums are extensions of geometric random sums. Up to the present, the negative–binomial random sums of independent identically distributed random variables have attracted much attention since they appear in many fields. However, for the strictly stationary m -dependent summands, such negative–binomial random sums are rarely studied. It seems that the complexity of the dependent structure has limited the use of classical tools based on independent random variables such as characteristic function. Up to now, a number of methods have been used to overcome the dependency such as the truncate method, Stein method, and Heinrich’s method. The paper deals with the order of approximation in weak limit theorems for normalized negative–binomial sums of strictly stationary m -dependent random variables generated by a sequence of independent, identically distributed random variables, using the moving average techniques. The orders of approximation in desired theorems are established in terms of the so-called Zolotarev ζ -metric. The obtained results are extensions and generalizations of several known results related to negative–binomial random sums and geometric random sums of independent, identically distributed random variables.