A class of strong laws for the sequences of nonnegative integer-valued random variables (Q1909432)

Let \(\{x_n\), \(n\geq 1\}\) be a sequence of random variables taking values in \(S= \{0,1, 2,\dots\}\) with the joint distribution \(f(x_1, \dots, x_n)\), \((p(0), p(1), \dots)\) be a probability distribution on \(S\) and \(r_n (\omega)= [\prod^n_{k=1} p(x_k) ]/ f(z_1, \dots, z_n)\) be the likelihood ratio. Also let \(m= \sum^\infty_{k=1} kp (k)<\infty\) and \(P(s)= \sum^\infty_{k=0} p(k) s^k\). The following strong limit theorems represented by inequalities are established: Theorem 1. Let \(c\geq 0\) be a constant, and \(D(c)= \{\omega: \liminf (1/n) \ln r_n (\omega)\geq -c\}\). Then \(\liminf (1/n) \sum^n_{k=1} x_k- m\geq \alpha(c)\) a.e. in \(D(c)\), where \(\alpha(c)= \sup\{ \varphi(s)\), \(0< s< 1\}\) if \(c>0\) and \(\alpha (0) =0\) and \(\varphi (s)= [\ln P(s)]/ \ln s- m+ c/\ln s\). Moreover, \(\alpha (c)\leq 0\), and \(\alpha (c)\to 0\) as \(c\to 0^+\). Theorem 2. Let \(R\) be the radius of convergence of the generating function \(P(s)\). If \(R>1\), then under the hypotheses of Theorem 1 we have \(\limsup (1/n) \sum^n_{k=1} x_k -m\leq \beta (c)\) a.e. in \(D(c)\), where \(\beta (c)= \inf\{ \varphi (s)\), \(1< s< R\}\) if \(c>0\) and \(\beta (0) =0\). Moreover, \(\beta (c)\geq 0\) and \(\beta (c)\to 0\) as \(c\to 0^+\). In the proof an approach of generating function in the study of strong limit theorems is proposed.