Theorems on large deviations for the sum of random number of summands

Abstract

In this paper, we present the rate of convergence of normal approximation and the theorem on large deviations for a compound process Zt = \sumNt i=1 t aiXi, where Z0 = 0 and ai > 0, of weighted independent identically distributed random variables Xi, i = 1, 2, . . . with mean EXi = µ and variance DXi = σ2 > 0. It is assumed that Nt is a non-negative integervalued random variable, which depends on t > 0 and is independent of Xi, i = 1, 2, . . . .

Large deviations for weighted random sums

“…(6) in two cases: µ = 0 and µ = 0 by pointing out the difference between them. It should be noted that large deviations when µ = 0 have been already considered in our papers [1,2], thus in this instance we pointed only some results without proofs.…”

“…Undoubtedly, there are a large amount of literature on theorems of large deviations for the r.s. under different assumptions and with various applications, for example, [1,2,[18][19][20][21][22][23][24]. Unfortunately, as far as we know, without our papers (see [1,2,21]) there are only few papers, e.g., [18,23,24], accordingly, by Aksomaitis (1965), Statulevičius (1967), Saulis and Deltuvienė (2007) on large deviations for the r.s.…”

“…under different assumptions and with various applications, for example, [1,2,[18][19][20][21][22][23][24]. Unfortunately, as far as we know, without our papers (see [1,2,21]) there are only few papers, e.g., [18,23,24], accordingly, by Aksomaitis (1965), Statulevičius (1967), Saulis and Deltuvienė (2007) on large deviations for the r.s. in the Cramér zone when all a j ≡ 1 and the cumulant method is used, although as it was already mentioned at the beginning of the Introduction, the cumulant method is an effective way of studying large deviation probabilities of various statistics.…”

“…In the present paper, we generalized theorems on large deviations for sums of nonrandom number of summands (see, e.g., [3,25]) and for sums of random number of summands presented in the papers [18,23,24], besides developed results presented in our papers [1,2,21] by investigating large deviation theorems both in the Cramér and power Linnik zones for weighted r.s. (1) in two cases: µ = 0, µ = 0.…”