Local theorems for the maximum of sums of independent identically distributed random variables

Aleshkyavichene, A. K.

doi:10.1007/bf01630724

Cited by 5 publications

(4 citation statements)

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“…In the case of finite variance the asymptotic behaviour of P(M n = x) has been studied by Aleshkyavichene [1] and by Nagaev and Eppel [10]. Moreover, in the case when Eχ + < ∞, (2) has been obtained by Alili and Doney [2].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

“…The authors have used it by proving an analog of (2) for random walks with Eχ + < ∞. It is worth mentioning that [1] contains another representation for P(M n = x), which is based on a recursive formula for the characteristic function of M n . The latter is due to Nagaev [9].…”

Section: Some Results From Fluctuation Theorymentioning

confidence: 99%

“…In this case we write X ∈ D (α, β). It is well known, that for every X ∈ D(α, β) there exists a sequence c n regularly varying of index 1/α such that {S [nt] /c n , t ∈ [0, 1]} converges weakly to the stable process {Y α,β (t) t ∈ [0, 1]} characterised by (1), i.e., Ee itY α,β (1) = G α,β (t).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Local limit theorem for the maximum of asymptotically stable random walks

Wachtel¹

2010

Probab. Theory Relat. Fields

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Let {S n ; n ≥ 0} be an asymptotically stable random walk and let M n denote it's maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for M n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal massfunction based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for M n does not hold.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 97%

Section: Some Results From Fluctuation Theorymentioning

confidence: 99%

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Local limit theorem for the maximum of asymptotically stable random walks

Wachtel¹

2010

Probab. Theory Relat. Fields

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“…A major ingredient in our proof will be the local limit theorem for maxima of sums of i.i.d. random variables from [Al2], see also [Al1,NE,Wa] for related results. To obtain (1.6) under minimal conditions, we need to extend the result from [Al2] from bounded to unbounded densities (see Proposition 4.2).…”

Section: Introductionmentioning

confidence: 97%