On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin

E. J. G. Pitman

doi:10.1017/S1446788700006121

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a real valued random variable with probability measure P and distribution function F. It will be convenient to take F as the intermediate distribution function defined by . In mathematical analysis it is a little more convenient to use this function rather than , which arise more naturally in probability theory. In all cases we shall consider With this definition, if the distribution function of X is F(x), then the distribution function of −X is 1−F(−x). The distribution of X is symmetrical about 0 if F(x) = 1 − F(−x).

References

[1]Pitman, E. J. G., ‘Some theorems on characteristic functions of probability distributions’. Proc. Fourth Berkeley Symp. Math. Statist. Prob. (1960) 2, 393–402Google Scholar

[2]Karamata, M. J., ‘Sur un mode de croissance régulière des fonctions’. Mathematica (Cluj), V. iv, (1930) 38–53.Google Scholar

[3]Karamata, M. J., ‘Sur un mode de croissance régulière.Théorèmes fondamentaux’. Bull. de la soc. math. de France. 61 (1933) 55–62.Google Scholar

[4]Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis (Cambridge University Press, 4th ed. 1927).Google Scholar

[5]Durell, C. V. and Robson, A., Advanced Trigonometry (G. Bell and Sons, 1930).Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Soni, K. and Soni, R. P. 1973. Lipschitz Behavior and Characteristic Functions. SIAM Journal on Mathematical Analysis, Vol. 4, Issue. 2, p. 302.

Ridenhour, J. R. and Soni, R. P. 1974. Parseval Relation and Tauberian Theorems for the Hankel Transform. SIAM Journal on Mathematical Analysis, Vol. 5, Issue. 5, p. 809.

Soni, Kusum and Soni, R.P. 1974. A tauberian theorem related to the modified Hankel transform. Bulletin of the Australian Mathematical Society, Vol. 11, Issue. 2, p. 167.

Bingham, N. H. and Doney, R. A. 1974. Asymptotic properties of supercritical branching processes I: The Galton-Watson process. Advances in Applied Probability, Vol. 6, Issue. 4, p. 711.

Soni, K and Soni, R.P 1975. Slowly varying functions and asymptotic behavior of a class of integral transforms I. Journal of Mathematical Analysis and Applications, Vol. 49, Issue. 1, p. 166.

Soni, K and Soni, R.P 1975. Slowly varying functions and asymptotic behavior of a class of integral transforms. II. Journal of Mathematical Analysis and Applications, Vol. 49, Issue. 2, p. 477.

Soni, K and Soni, R.P 1975. Slowly varying functions and asymptotic behavior of a class of integral transforms III. Journal of Mathematical Analysis and Applications, Vol. 49, Issue. 3, p. 612.

Emery, D. J. 1975. On a condition satisfied by certain random walks. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 31, Issue. 2, p. 125.

Luther, Wolfram 1977. �ber das asymptotische Verhalten der trigonometrischen und Hankelschen Integraltransformation. Archiv der Mathematik, Vol. 29, Issue. 1, p. 614.

Haan, Laurens De 1977. On functions derived from regularly varying functions. Journal of the Australian Mathematical Society, Vol. 23, Issue. 4, p. 431.

Miroshin, R. N. 1978. Conditions of Local Nondeterminism of Differentiable Gaussian Stationary Processes. Theory of Probability & Its Applications, Vol. 22, Issue. 4, p. 831.

Hall, Peter 1978. On the rate of convergence of moments in the central limit theorem. Journal of the Australian Mathematical Society, Vol. 25, Issue. 2, p. 250.

Hall, Peter 1979. On the rate of convergence in the central limit theorem for distributions with regularly varying tails. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 49, Issue. 1, p. 1.

Malevich, T. L. 1980. On Conditions for the Moments of the Number of Zeros of Gaussian Stationary Processes to Be Finite. Theory of Probability & Its Applications, Vol. 24, Issue. 4, p. 741.

Frenk, J. B. G. 1982. The behavior of the renewal sequence in case the tail of the waiting-time distribution is regularly varying with index −1. Advances in Applied Probability, Vol. 14, Issue. 4, p. 870.

Pitman, E. J. G. 1982. The Making of Statisticians. p. 111.

Doney, R.A. 1982. On the exact asymptotic behaviour of the distribution of ladder epochs. Stochastic Processes and their Applications, Vol. 12, Issue. 2, p. 203.

Malevich, T. L. 1985. On Conditions for Finiteness of the Factorial Moments of the Number of Zeros of Gaussian Stationary Processes. Theory of Probability & Its Applications, Vol. 29, Issue. 3, p. 534.

Welsh, A. H. 1986. ON THE USE OF THE EMPIRICAL DISTRIBUTION AND CHARACTERISTIC FUNCTION TO ESTIMATE PARAMETERS OF REGULAR VARIATION. Australian Journal of Statistics, Vol. 28, Issue. 2, p. 173.

Download full list

Article contents

On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin

Extract

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests