Nonlinear perturbed renewal equations with application to a random walk (Q2732665)

Let NEWLINE\[NEWLINEx_\varepsilon(n)=q_\varepsilon(n)+\sum_{t=0}^n x_\varepsilon(n-t)f_\varepsilon(t), \quad n \geq 0,NEWLINE\]NEWLINE be a discrete time renewal equation, where the free term \(q_\varepsilon(n)\) and the distribution \(f_\varepsilon(t)\) depend on some perturbation parameter \(\varepsilon \geq 0\). It is assumed that \(f_\varepsilon(n)\to f_0(n)\) and \(q_\varepsilon(n)\to q_0(n)\) as \(\varepsilon\to 0\). It is also assumed that the distribution \(f_\varepsilon(n)\) can be improper, which means that the defect \(f_\varepsilon=1-\sum_{n=0}^\infty f_\varepsilon(n) \leq 1\) for \(\varepsilon>0\), but \(f_\varepsilon\to f_0=0\) as \(\varepsilon \to 0\). The author is interested in asymptotics of the form \(x_\varepsilon(n_\varepsilon)/ e^{-\rho_\varepsilon n_\varepsilon} \to x_0(\infty)\) as \(\varepsilon\to 0\), where \(n_\varepsilon\to\infty\) and the normalizing coefficient \(\rho_\varepsilon \to 0\) as \(\varepsilon\to 0\).NEWLINENEWLINENEWLINEThese asymptotics can be considered as a generalization of the well-known discrete time renewal theorem [see \textit{W. Feller}, ``An introduction to probability theory and its applications'' (1950; Zbl 0039.13201)]. These types of asymptotics were first investigated by \textit{D. S. Sil'vestrov} [Dopov. Akad. Nauk Ukr. RSR, Ser. A 1976, 978-981 (1976; Zbl 0357.60009), Teor. Veroyatn. Mat. Stat. 18, 144-164 (1978; Zbl 0415.60080) and ibid. 20, 97-117 (1979; Zbl 0431.60084)] and by \textit{V. M. Shurenkov} [Mat. Sb., Nov. Ser. 112(154), 115-132 (1980; Zbl 0434.60090) and ibid. 112(154), 226-241 (1980; Zbl 0444.60071)]; \textit{D. Alimov} and \textit{V. M. Shurenkov} [Ukr. Math. J. 42, No. 11, 1283-1288 (1990); translation from Ukr. Mat. Zh. 42, No. 11, 1443-1448 (1990; Zbl 0719.60070)]; \textit{D. Alimov} [Theory Probab. Appl. 39, No. 4, 537-546 (1994); translation from Teor. Veroyatn. Primen. 39, No. 4, 657-668 (1994; Zbl 0840.60060)]; \textit{M. Gyllenberg} and \textit{D. S. Silvestrov} [J. Math. Biol. 33, No. 1, 35-70 (1994; Zbl 0816.92016), Theory Stoch. Process. 5(21), No. 1-2, 6-21 (1999; Zbl 0952.60080), Stochastic Processes Appl. 86, No. 1, 1-27 (2000), Insur. Math. Econ. 26, No. 1, 75-90 (2000; Zbl 0956.91044)]; \textit{Ya. I. Jelejko} [Theory Probab. Math. Stat. 52, 69-74 (1996); translation from Teor. Jmovirn. Mat. Stat. 52, 66-71 (1995; Zbl 0948.60078)]; \textit{D. S. Silvestrov} [ibid. 52, 153-162 (1996), resp. ibid. 52, 143-153 (1995; Zbl 0946.60080)]; the author [ibid. 60, 35-42 (2000), resp. ibid. 60, 31-37 (1999; Zbl 0955.60080) and ibid. 61, 21-32 (2000), resp. ibid. 61, 21-32 (2000)]. NEWLINENEWLINENEWLINE The author and \textit{D. S. Silvestrov} [Teor. Sluch. Protsess. 3(19), No. 1-2, 164-176 (1997; Zbl 0946.60079)] improved the asymptotic. It was assumed that the defect \(f_\varepsilon=1- \sum_{n=0}^\infty f_\varepsilon(n)\) and the moments \(m_{\varepsilon r}=\sum_{n=0}^\infty n^r f_\varepsilon(n),\) \( r=1,\dots,k\), could be expanded in asymptotical polynomial series, \( f_\varepsilon=g_1\varepsilon+ \cdots+g_k\varepsilon^k+o(\varepsilon^k)\) as \(\varepsilon\to 0\) and \(m_{\varepsilon r}=d_{0 r}+d_{1 r}\varepsilon+ \cdots+d_{k r}\varepsilon^k+o(\varepsilon^k)\) as \(\varepsilon \to 0\). These relations can be interpreted as nonlinear polynomial type perturbation conditions for the corresponding functionals of the distribution \(f_\varepsilon(n)\). It was also assumed that the condition, where the rate of perturbation is balanced with the rate of time growth, \(\varepsilon^zn_\varepsilon \to\lambda_z\) as \(\varepsilon\to 0\), where \(1 \leq z \leq k\) is integer and \(n_\varepsilon\to \infty\) as \(\varepsilon \to 0\). Under these conditions the asymptotic was improved to the following more explicit form NEWLINE\[NEWLINE{x_\varepsilon(n_\varepsilon) \over \exp\{-(b_1\varepsilon+ \cdots +b_{z-1}\varepsilon^{z-1})n_\varepsilon\}} \to e^{-b_z\lambda_z}x_0(\infty) \quad \text{as } \varepsilon \to 0.NEWLINE\]NEWLINE This asymptotic provides a tool for studying the mixed ergodic and large deviation theorems for the perturbed renewal equations. It could also be used in theorems for the analysis of the so-called quasi-stationary phenomena for queueing systems, populations dynamics modes, etc. NEWLINENEWLINENEWLINEIn this paper the author continues this line of research. The mixed polynomial-exponential perturbation conditions are used for the defect and the first moment NEWLINE\[NEWLINEf_\varepsilon=(g_k\varepsilon^k+\cdots +g_l\varepsilon^l)e^{- a/\varepsilon}+(h_{k'}\varepsilon^{k'}+\cdots +h_{l'}\varepsilon^{l'})e^{-2a/\varepsilon} +o(\varepsilon^{l'}e^{-2a/\varepsilon}),NEWLINE\]NEWLINE NEWLINE\[NEWLINEm_{\varepsilon 1}=d_0+(d_u'\varepsilon^u+ \cdots + d_v'\varepsilon^v)e^{-a/\varepsilon}+o(\varepsilon^ve^{-a/\varepsilon}) NEWLINE\]NEWLINE and the polynomial form for the second moment in the form NEWLINE\[NEWLINE m_{\varepsilon 2}=e_0+e_1\varepsilon+\cdots+e_w\varepsilon^w+o(\varepsilon^w).NEWLINE\]NEWLINE The balancing condition in this case takes the form \(\varepsilon^ze^{-a/\varepsilon}n_\varepsilon \to\lambda_z\) as \(\varepsilon \to 0\), where \(k \leq z \leq l\), or \(\varepsilon^ze^{-2a/\varepsilon}n_\varepsilon \to \mu_z\) as \(\varepsilon \to 0\). NEWLINENEWLINENEWLINEUnder the perturbation conditions the following asymptotic expansion for \(\rho_\varepsilon\) holds true NEWLINE\[NEWLINE\rho_\varepsilon=(b_k\varepsilon^k+ \cdots +b_l\varepsilon^l)e^{-a/\varepsilon}+(c_m\varepsilon^m+ \cdots +c_n\varepsilon^n)e^{-2a/\varepsilon}+o(\varepsilon^ne^{- 2a/\varepsilon}).NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThen the asymptotic can be improved by use of one of the balancing conditions and the asymptotical expansion for \(\rho_\varepsilon\). For example, the asymptotic can be improved to the following form NEWLINE\[NEWLINE{x_\varepsilon(n_\varepsilon) \over \exp\{-((b_k\varepsilon^k+ \cdots +b_l\varepsilon^l)e^{-a/\varepsilon}+(c_m\varepsilon^m+ \cdots +c_{z-1}\varepsilon^{z-1})e^{-2a/\varepsilon})n_\varepsilon\}} \rightarrow e^{-c_z\mu_z}x_0(\infty)NEWLINE\]NEWLINE as \(\varepsilon \to 0.\) The author studies the exponential asymptotic for the perturbed renewal equations with mixed polynomial exponential perturbations as described above, i.e. involving polynomial combined with two different exponential terms. The results are applied to a Bernoullian random walk on a positive half-line with moving barrier.