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In this work, we propose an accreting stellar binary model for understanding the active periodic fast radio bursts (FRBs). The system consists of a stellar compact object (CO) and a donor star (DS) companion in an eccentric orbit, where the DS fills its own Roche lobe near the periastron. The CO accretes the material from the DS and then drive relativistic magnetic blobs. The interaction between the magnetic blobs and the stellar wind of the DS produces a pair of shocks. We find that both of the reverse shock and the forward shock are likely to produce FRBs via synchrotron maser mechanism. We show that this system can in principle sufficiently produce highly active FRBs with a long lifetime, and also can naturally explain the periodicity and the duty cycle of the activity as appeared in FRBs 180916 and 121102. The radio nebula excited by the long-term injection of magnetic blobs into the surrounding environment may account for the associated persistent radio source. In addiction, we discuss the possible multi-wavelength counterparts of FRB 180916 in the context of this model. Finally, we encourage the search for FRBs in the ultraluminous X-ray sources., 11 pages, 2 figures; accepted to APJ

The cosmic evolution of gamma-ray burst (GRB) luminosity is essential for revealing the GRB physics and for using GRBs as cosmological probes. We investigate the luminosity evolution of long GRBs with a large sample of 258 {\em Swift}/BAT GRBs. Parameterized the peak luminosity of individual GRBs evolves as $L_{\rm p}\propto{\rm }(1+z)^{k}$, we get $k=1.49\pm0.19$ using the non-parametric $\tau$ statistics method without considering observational biases of GRB trigger and redshift measurement. By modeling these biases with the observed peak flux and characterizing the peak luminosity function of long GRBs as a smoothly broken power-law with a break that evolves as $L_{\rm b}\propto (1+z)^{k_{\rm b}}$, we obtain $k_{\rm b}=1.14^{+0.99}_{-0.47}$ through simulations based on assumption that the long GRB rate follows the star formation rate (SFR) incorporating with cosmic metallicity history. The derived $k$ and $k_b$ values are systematically smaller than that reported in previous papers. By removing the observational biases of the GRB trigger and redshift measurement based on our simulation analysis, we generate mock {\em complete} samples of 258 and 1000 GRBs to examine how these biases affects on the $\tau$ statistics method. We get $k=0.94\pm 0.14$ and $k=0.80\pm 0.09$ for the two samples, indicating that these observational biases may lead to overestimate the $k$ value. With the large uncertain of $k_b$ derived from our simulation analysis, one even cannot convincingly argue a robust evolution feature of the GRB luminosity., Comment: 37 pages, 6 figures, accepted for publication in ApJ