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The purpose of this paper is to study a chi-square-type distribution who degrees of freedom are geometric random variables in connection with weak limiting distributions of geometric random sums of squares of independent, standard normal distributed random variables. Some characteristics of chi-square-type random variables with geometrically distributed degrees of freedom including probability density function, probability distribution function, mean and variance are calculated. Some asymptotic behaviors of chi-square-type random variables with geometrically distributed degrees of freedom are also established via weak limit theorems for normalized geometric random sums of squares of independent, standard normal distributed random variables. The rates of convergence in desired weak limit theorems also estimated through Trotter’s distance. The received results are extensions and generalizations of several known results.

A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums