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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:
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[2] 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
[3] 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
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