Strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q× 1 responses, bounded and fixed p × q regressors Zi and general link function, under the most general assumption on the minimum eigenvalue of\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\nolimits_{i = 1}^n {Z_i Z'_i } $$ \end{document} the moment condition on responses as weak as possible and other mild regular conditions, we prove that with probability one, the quasi-likelihood equation has a solution\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat \beta $$ \end{document}n for all large sample size n, which converges to the true regression parameter β0. This result is an essential improvement over the relevant results in literature.