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On the problem of characterizing probability distributions by the absolute moments of partial sums. (English. Russian original) Zbl 0915.60021

Theory Probab. Appl. 42, No. 3, 426-432 (1997); translation from Teor. Veroyatn. Primen. 42, No. 3, 473-481 (1997).
For \(j=1,2\) let \(X_{1j}, X_{2j},\dots\) be a sequence of i.i.d. integrable random variables, let \(F_j\) denote the distribution of \(X_{1j}\), let \(\varphi_j\) stand for the characteristic function of \(X_{1j}\), and put \(a_{nj}= E| \sum^n_{k=1} X_{kj}|\), \(n\geq 1\). The authors are interested in finding sufficient and close to necessary conditions under which \(a_{n1}=a_{n2}\) for each \(n\geq 1\) implies \(F_1=F_2\). In this connection they prove the next result: For \(j=1,2\) assume that Cramer’s condition \(\lim\sup_{t\to \infty} | \varphi_j(t) |<1\) is satisfied, and assume that there exists \(b_j>0\) such that \(\text{Im} \varphi_j (t)=0\), \(-b_j\leq t\leq b_j\), while \(\varphi_j\) strictly decreases on \([0,b_j]\). Then \(a_{n1}=a_{n2}\), \(n\geq 1\), implies that \(\varphi_1= \varphi_2\) on some interval \([-\beta,\beta]\).

MSC:

60E05 Probability distributions: general theory
60E10 Characteristic functions; other transforms
60G50 Sums of independent random variables; random walks
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