Asymptotic expansions for probability distributions. II
Copyright policy )Asymptotic expansions for probability distributions. II
[For part III see the foregoing review; Zbl 0571.60033.] Let \(\xi\) be a lattice random variable on \(\{\) 0,1,2,...\(\}\) with the probability distribution p(k). It is assumed that the cumulants \(\gamma_{\ell}\) of order \(\ell =1,2,..\). satisfy the conditions: \(\gamma_ 1>0,| \gamma_{\ell}| \leq Hc^{\ell}\ell !/\Delta^{\ell -1}\), \(\ell =2,3,..\).. Here is \(\Delta >0\) with some increasing parameters, \(H=H(\Delta)>0\), \(c=c(\Delta)>0\)- with limited functions. Using the cumulant technique asymptotic expansions are obtained: for p(k) - in part I [ibid. 23, No.3, 196-213 (1983; Zbl 0536.60033)], and for \(P\{\xi
integral limit theorems, nonnegative integer-valued random variables, Local limit theorems for a lattice variable, Local theorems for continuous random variables, Bernstein-Statulevičius type condition, lattice random variable, numerical estimations, Central limit and other weak theorems, asymptotic expansions, cumulant technique, Limit theorems in probability theory, Asymptotic expansions, cumulants, Integral limit theorems, expansions
integral limit theorems, nonnegative integer-valued random variables, Local limit theorems for a lattice variable, Local theorems for continuous random variables, Bernstein-Statulevičius type condition, lattice random variable, numerical estimations, Central limit and other weak theorems, asymptotic expansions, cumulant technique, Limit theorems in probability theory, Asymptotic expansions, cumulants, Integral limit theorems, expansions
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