Abstract:
We obtain some integro-local and integral limit theorems for the sums S(n)=ξ(1)+⋯+ξ(n) of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form P(ξ⩾t)=e−tβL(t), where β∈(0,1) and L(t) is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as x→∞ of the probabilities
P(S(n)∈[x,x+Δ)) and P(S(n)⩾x)
in the zone of normal deviations and all zones of large deviations of x: in the Cramer and intermediate zones, and also in the “extreme” zone where the distribution of S(n) is approximated by that of the maximal summand.
Keywords:
semiexponential distribution, integro-local theorem, Cramér series, segment of the Cramér series, random walk, large deviations, Cramér zone of deviations, intermediate zone of deviations, zone of approximation by the maximal summand.
Citation:
A. A. Borovkov, A. A. Mogul'skii, “Integro-local and integral theorems for sums of random variables with semiexponential distributions”, Sibirsk. Mat. Zh., 47:6 (2006), 1218–1257; Siberian Math. J., 47:6 (2006), 990–1026
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\by A.~A.~Borovkov, A.~A.~Mogul'skii
\paper Integro-local and integral theorems for sums of random variables with semiexponential distributions
\jour Sibirsk. Mat. Zh.
\yr 2006
\vol 47
\issue 6
\pages 1218--1257
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\jour Siberian Math. J.
\yr 2006
\vol 47
\issue 6
\pages 990--1026
\crossref{https://doi.org/10.1007/s11202-006-0110-x}
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Linking options:
https://www.mathnet.ru/eng/smj930
https://www.mathnet.ru/eng/smj/v47/i6/p1218
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