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Consider an $(L,1)$ random walk in an i.i.d. random environment, whoseenvironment involves certain parameter. We get the maximum likelihoodestimator(MLE) of the environment parameter which can be written as functionalsof a multitype branching process with immigration in a randomenvironment(BPIRE). Because the offspring distributions of the involvedmultitype BPIRE are of the linear fractional type, the limit invariantdistribution of the multitype BPIRE can be computed explicitly. As a result, weget the consistency of the MLE. Our result is a generalization of Comets et al.[Stochastic Process. Appl. 2014, 124, 268-288].

Maximum likelihood estimator and its consistency for an $(L,1)$ random walk in a parametric random environment