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We study the law of the iterated logarithm (LIL) for the maximum likelihoodestimation of the parameters (as a convex optimization problem) in thegeneralized linear models with independent or weakly dependent ($\rho$-mixing,$m$-dependent) responses under mild conditions. The LIL is useful to derive theasymptotic bounds for the discrepancy between the empirical process of thelog-likelihood function and the true log-likelihood. As the application of theLIL, the strong consistency of some penalized likelihood based model selectioncriteria can be shown. Under some regularity conditions, the model selectioncriterion will be helpful to select the simplest correct model almost surelywhen the penalty term increases with model dimension and the penalty term hasan order higher than $O({\rm{loglog}}n)$ but lower than $O(n)$. Simulationstudies are implemented to verify the selection consistency of BIC.

Law of the Iterated Logarithm and Model Selection Consistency for GLMs with Independent and Dependent Responses