Some properties of subexponential distributions

A. L. Yakymiv

Matematičeskie zametki, Tome 62 (1997) no. 1, pp. 138-144.

Résumé

The nonnegative random variable $X$ is said to have a subexponential distribution if we have $\bigl(1-G(t)\bigr)\big/\bigl(1-F(t)\bigr)\to2$ as $t\to\infty$, where $F(t)=\mathsf P\{X\le t\}$ and $G(t)$ is the convolution of $F(t)$ with itself. Conditions on the distribution of independent nonnegative random variables $X$ and $Y$ such that $\max(X,Y)$ and $\min(X,Y)$ have a subexponential distribution are given.

Bibliographie

[1] Chistyakov V. P., “Teorema o summakh nezavisimykh polozhitelnykh sluchainykh velichin i ee prilozheniya k vetvyaschimsya sluchainym protsessam”, Teor. veroyatnostei i ee primeneniya, 9:4 (1964), 710–718 | MR | Zbl

[2] Sgibnev M. S., “Asimptotika bezgranichno delimykh raspredelenii v $\mathbb R$”, Sib. matem. zh., 31:1 (1990), 135–140 | MR

[3] Chover J., Ney P., Wainger S., “Degeneracy properties of subcritical branching processes”, Ann. Probab., 1 (1973), 663–673 | DOI | MR | Zbl

[4] Teugels J. L., “The class of subexponential distributions”, Ann. Probab., 3 (1975), 1000–1011 | DOI | MR | Zbl

[5] Goldie C. M., “Subexponential distributions and dominated-variation tails”, J. Appl. Probab., 15 (1978), 440–442 | DOI | MR | Zbl

[6] Embrechts P., Goldie C. M., “On closure and factorisation properties of subexponential and related distributions”, J. Austral. Math. Soc., 29 (1980), 243–256 | DOI | MR | Zbl

[7] Cline D. B. H., “Convolution of distributions with exponential and subexponential tails”, J. Austral. Math. Soc., 93 (1987), 347–365 | DOI | MR

[8] Kluppelberg C., “Subexponential distributions and integrated tails”, J. Appl. Probab., 25 (1988), 132–141 | DOI | MR

[9] Leslie J. R., “On the non-closure under convolution of the subexponential family”, J. Appl. Probab., 26 (1989), 58–66 | DOI | MR | Zbl

[10] Yakymiv A. L., “Dostatochnye usloviya subeksponentsialnosti svertki dvukh raspredelenii”, Matem. zametki, 58:5 (1995), 778–781 | MR | Zbl