ner_tags: O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O
tokens: other words, the PDFs are equal, because the system self-adjusts, not because glitch sizes are correlated with forward waiting times, as in the α ≪ α c (β) regime in Section 4 . The explanation for the qualitative behaviour seen in Fig. 20 is the same as for Fig. 13 . Figure 20. Correlation between sizes and forward/backward waiting times: Pearson coefficient, r , as a function of the control parameter α, for separable η with φ( x ) = x δ . For each value of α, we sample Δ t and Δ X over the course of a single simulation of N = 10 6 glitches, after the system reaches stationarity. We then calculate the pairwise correlation between the two sets for each α. Parameter: δ = 7. 6.5 Lag-dependent spin-down For the sake of completeness, we extend the results of Section 6 to the case of a general lag-dependent spin-down torque in Appendix E . 7 SAMPLING SIZE BIAS We now consider more carefully how we sample the different random variables used throughout this paper: X , N , Δ t , ΔΩ c (∝Δ X ), X e , and X s . Both X and N differ fundamentally from the other variables, because they are defined continuously for all t . The other variables are defined at discrete instants, when N ( t ) jumps by one due to a glitch. We therefore sample X ( t ) and N ( t ) by running an ensemble of simulations and observe each simulation at some time t 0 . The other variables are sampled at every glitch in a single simulation once it reaches stationarity. As an example of the subtle consequences of sampling bias, consider the ordering of the first moments 〈 X 〉, 〈 X s 〉, and 〈 X e 〉. Fig. 21 shows p ( x ), p e ( x ), and p s ( x ) for the non-separable power-law η ( 18 ) and ( 19 ) from Section 4 . Note that p s ( x ) is skewed towards larger values than p ( x ). By integrating the curves, we find 〈 X s 〉 = 0.600, 〈 X 〉 = 0.579, 〈 X e 〉 = 0.667, and hence 〈 X 〉 〈 X s 〉 〈 X e 〉. This seems wrong: X s , the value of X immediately after a glitch, ought to be smaller, on average, than X measured at some arbitrary time t . However, X is sampled from an ensemble at fixed time t 0 , whereas X s and X e are drawn from a different ensemble (every glitch event in a single simulation). If we observe the system at time t , we are more likely to land in a longer inter-glitch time interval than a smaller one, in which the (biased) X s and X e are spaced farther apart. Figure 21. Stationary PDFs p s ( x ), p e ( x ), and p ( x ) for the λ and power-law η used throughout Section 4 . Both p e ( x ) and p s ( x ) are sampled from a single simulation of N = 10 8 glitches after stationarity. We obtain p ( x ) by sampling an ensemble of S = 10 6 simulations at t = 8. Parameters: β = 10 −2 and α = 5. Sampling bias has an important observable consequence, even though p s and p e cannot be observed. We can obtain an ensemble distribution of waiting times by choosing a time t 0 ≫ 1 and measuring the length of the interval T ( i + 1) − T ( i ) for every simulation in the ensemble, where T ( i ) is