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On sequential maxima of exponential sample means, with an application to ruin probability. (English) Zbl 1448.60031
Consider a sequence $$(X_{i})_{i\geq 1}$$ of independent identically distributed (i.i.d.) random variables, each having exponential distribution with mean 1. For each $$i\in \mathbb{N}^{*}$$, define the sample mean of the first $$i$$ variables as $$\overline{X_{i}}:=\frac{X_1+X_2+\dots+X_{i}}{i}$$ and the supremum of this sequence as $Z_{\infty}:=\sup\{\overline{X_{i}}:i\in \mathbb{N}^{*}\}.$ In this note the authors compute the distribution function, $$F_{\infty}$$, of $$Z_{\infty}$$ and with some restriction on $$F_{\infty}$$, they give the inverse of this distribution function. More precisely, they obtain the following main result:
Theorem 1.
(a)
$$Z_{\infty}$$ has distribution function $F_{\infty}(x)=1-\sum_{k=1}^\infty\frac{k^{k-1}}{k!}x^{k-1}e^{-kx}$ for $$x>0$$, and density which is continuous on $$\mathbb{R}\backslash\{1\}$$, positive on $$]1,+\infty[$$, and zero on $$]-\infty, 1[$$.
(b)
The restriction of $$F_{\infty}$$ on $$]1, +\infty[$$ is one to one and onto $$]0, 1[$$ with inverse $F_{\infty}^{-1}(u)=-\frac{\log(1-u)}{u}\text{ for all }u\in]0, 1[$

As application to ruin probability, the authors consider the following risk model defined by $U_{n}=u+cn-S_{n},\quad n\in \mathbb{N}^{*},$ where they assume that the aggregate claim at time $$n$$ is described by $$S_{n}:=X_1+\dots+X_{n}$$, the $$(X_{i})_{i\geq 1}$$ are i.i.d. with $$\mathbb{E}(X_1)=1$$, the premium rate (per time unit) is $$c=1+\theta>0$$ ($$\theta$$ is the safety loading of the insurance), and the initial capital is $$u>-(1+\theta)$$, where negative initial capital is allowed for technical reasons. The ruin probability $\psi(u):=Pr(Un<0\text{ for some }n\in \mathbb{N}^{*})\tag{$$\ast$$}$ is of fundamental importance in this application.
This particular problem (for general claims) has been studied in [P. Sattayatham et al., “Ruin probability-based initial capital of the discrete-time surplus process”, Variance Adv. Sci. Risk 7, No. 1, 74–81 (2013)], while the probability of ruin for more general models is studied in detail in [S. Asmussen and H. Albrecher, Ruin probabilities. 2nd ed. Hackensack, NJ: World Scientific (2010; Zbl 1247.91080)] and for other applications we can see [R. M. Corless et al., Adv. Comput. Math. 5, No. 4, 329–359 (1996; Zbl 0863.65008)].
The authors obtain the following result as application to ruin probability:
Theorem 2. Assume that the i.i.d. individual claims $$(X_{i})_{i\geq 1}$$ are exponential random variables with mean 1, fix $$\alpha\in]0, 1[$$ and $$\theta>0$$, and set $$c=1+\theta$$. Then,
(a)
the ruin probability $$(\ast)$$ is given by $\psi(u)=\begin{cases} \frac{t(c)}{c}\exp(-u(1-\frac{t(c)}{c}))&\text{ if }u>-c, \\ 1&\text{ if }u\leq c, \end{cases}$ where the function $$t$$ is given by $t(x):=g_{r}^{-1}(xe^{-x})=h(xe^{-x}),\quad x\geq 0\text{ and }g_{r}(t)=g_{r}(t(x))=xe^{-x};$
(b)
the minimum initial capital $$u=u(\alpha,\theta)$$ needed to ensure that $$\Psi(u)\leq\alpha$$ is given by the unique root of the equation $(1+\theta+u)(1-\alpha^{\frac{1+\theta}{1+\theta+u}})=-\log\alpha,\quad u>-(1+\theta).$
##### MSC:
 60E05 Probability distributions: general theory 91G05 Actuarial mathematics 60G70 Extreme value theory; extremal stochastic processes
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##### References:
  Asmussen, S. and Albrecher, H.: Ruin probabilities. Vol. 14. Singapore: World Scientific, 2010. · Zbl 1247.91080  Charalambides, C.: Abel series distributions with applications to fluctuations of sample functions of stochastic processes. Communications in Statistics - Theory & Methods, 19(1), (1990), 317-335. · Zbl 0900.62069  Corless, R.M, Gonnet G.H., Hare, D.E.G., Jeffrey D.J., and Knuth D.E.: On the Lambert $$W$$ function. Advances in Computational Mathematics, 5(1), (1996), 329-359. · Zbl 0863.65008  Csörgő, M. and Révész, P.: Strong approximations in probability and statistics. Academic Press, 1981. · Zbl 0539.60029  Durrett, R.: Probability: theory and examples. 4th Edition. Cambridge University Press, 2010. · Zbl 1202.60001  Sattayatham, P., Sangaroon, K., and Klongdee, W.: Ruin probability-based initial capital of the discrete-time surplus process. Variance, Advancing the Science of Risk, 7(1), (2013), 74-81.  Stanley, R. and Pitman, J.: A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete & Computational Geometry, 27(4), (2002), 603-634. · Zbl 1012.52019
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