Some results on the joint distribution of the renewal Epochs prior to a given time instant

Schulte-Geers, Ernst; Stadje, Wolfgang

doi:10.1016/0304-4149(88)90077-4

Cited by 7 publications

(2 citation statements)

References 11 publications

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“…, X n given S n can be found in [7], [8], [10], and [13]. A functional limit theorem under the condition S n = constant is given in [14], in [18] a convergence result for expectations conditioned on a sum is proved, and in [2] an approximation for the distribution of (X 1 , . .…”

Section: Introductionmentioning

confidence: 98%

Conditional Limit Theorems for the Terms of a Random Walk Revisited

Bar‐Lev

Schulte-Geers

Stadje

2013

Journal of Applied Probability

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In this paper we derive limit theorems for the conditional distribution ofX1givenSn=snasn→ ∞, where theXiare independent and identically distributed (i.i.d.) random variables,Sn=X1+··· +Xn, andsn/nconverges orsn≡sis constant. We obtain convergence in total variation of PX1∣Sn/n=sto a distribution associated to that ofX1and of PnX1∣Sn=sto a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

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Section: Introductionmentioning

confidence: 98%

Conditional Limit Theorems for the Terms of a Random Walk Revisited

Bar‐Lev

Schulte-Geers

Stadje

2013

Journal of Applied Probability

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“…Asymptotic results for linear combinations of X 1 , . A functional limit theorem under the condition S n = constant is given in [14], in [18] a convergence result for expectations conditioned on a sum is proved, and in [2] an approximation for the distribution of (X 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Conditional Limit Theorems for the Terms of a Random Walk Revisited

Bar‐Lev¹,

Schulte-Geers²,

Stadje³

2013

J. Appl. Probab.

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In this paper we derive limit theorems for the conditional distribution of X 1 given S n = s n as n → ∞, where the X i are independent and identically distributed (i.i.d.) random variables, S n = X 1 + · · · + X n , and s n /n converges or s n ≡ s is constant. We obtain convergence in total variation of P X 1 | S n /n=s to a distribution associated to that of X 1 and of P nX 1 | S n =s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

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Inspection Paradox (Update)

Kamps¹

2004

Encyclopedia of Statistical Sciences

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Some results on the joint distribution of the renewal Epochs prior to a given time instant

Cited by 7 publications

References 11 publications

Conditional Limit Theorems for the Terms of a Random Walk Revisited

Conditional Limit Theorems for the Terms of a Random Walk Revisited

Conditional Limit Theorems for the Terms of a Random Walk Revisited

Inspection Paradox (Update)

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