On the almost sure invariance principle for dependent Bernoulli random variables

We consider a sequence of dependent Bernoulli variables where the success probability of the trial conditional on the past history is a linear function of the mean number of successes achieved to that point. An almost sure invariance principle is established for the partial sum and we also generalize the model to a multi-dimensional case, extending the results of Heyde (2004), James et al. (2008) and Wu et al. (2012).