## Bernoulli

• Bernoulli
• Volume 17, Number 3 (2011), 942-968.

### Functional limit theorems for sums of independent geometric Lévy processes

Zakhar Kabluchko

#### Abstract

Let $ξ_i, i ∈ ℕ$, be independent copies of a Lévy process {$ξ(t), t ≥ 0$}. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process $$Z_N(t)=\sum_{i=1}^N \mathrm{e}^{\xi_i(s_N+t)}$$ as $N → ∞$, where $s_N$ is a non-negative sequence converging to $+∞$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\lim \inf _{N→∞ } \log N / s_N > λ_2$ for some critical value $λ_2 > 0$, then the limit is an Ornstein–Uhlenbeck process. However, if $λ := \lim _{N→∞ }\log N /s_N ∈ (0, λ_2)$, then the limit is a certain completely asymmetric $α$-stable process $\mathbb{Y}_{\alpha ;\xi}$.

#### Article information

Source
Bernoulli, Volume 17, Number 3 (2011), 942-968.

Dates
First available in Project Euclid: 7 July 2011

https://projecteuclid.org/euclid.bj/1310042851

Digital Object Identifier
doi:10.3150/10-BEJ299

Mathematical Reviews number (MathSciNet)
MR2817612

Zentralblatt MATH identifier
1304.00020

#### Citation

Kabluchko, Zakhar. Functional limit theorems for sums of independent geometric Lévy processes. Bernoulli 17 (2011), no. 3, 942--968. doi:10.3150/10-BEJ299. https://projecteuclid.org/euclid.bj/1310042851

#### References

• [1] Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. New York: Wiley.
• [2] Bahadur, R. and Ranga Rao, R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015–1027.
• [3] Ben Arous, G., Bogachev, L. and Molchanov, S. (2005). Limit theorems for sums of random exponentials. Probab. Theory Related Fields 132 579–612.
• [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Chichester: Wiley.
• [5] Bogachev, L. (2006). Limit laws for norms of IID samples with Weibull tails. J. Theoret. Probab. 19 849–873.
• [6] Bogachev, L. (2007). Extreme value theory for random exponentials. In Probability and Mathematical Physics. A Volume in Honor of Stanislav Molchanov (D. Dawson et al., eds.). CRM Proceedings and Lecture Notes 42 41–64. Providence, RI: Amer. Math. Soc.
• [7] Bovier, A., Kurkova, I. and Löwe, M. (2002). Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30 605–651.
• [8] Cranston, M. and Molchanov, S. (2005). Limit laws for sums of products of exponentials of i.i.d. random variables. Israel J. Math. 148 115–136.
• [9] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics 38. New York: Springer.
• [10] Hahn, M.G. (1978). Central limit theorems in D[0, 1]. Z. Wahrsch. Verw. Gebiete 44 89–101.
• [11] Ibragimov, R. and Sharakhmetov, S. (2002). On extremal problems and best constants in moment inequalities. Sankhyā A 64 42–56.
• [12] Janßen, A. (2010). Limit laws for power sums and norms of i.i.d. samples. Probab. Theory Related Fields 146 515–533.
• [13] Kabluchko, Z. (2009). Functional limit theorems for sums of independent geometric Lévy processes. Preprint version of the present paper. Available at http://arxiv.org/abs/0911.4139v1.
• [14] Kabluchko, Z. (2009). Limiting distributions for sums of independent random products. Not published. Available at http://arxiv.org/abs/0904.4127.
• [15] Kabluchko, Z. (2010). Limit laws for sums of independent random products: The lattice case. J. Theoret. Probab. To appear. Available at http://arxiv.org/abs/1003.1657.
• [16] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624–632.
• [17] Meerschaert, M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. New York: Wiley.
• [18] Petrov, V. (1965). On the probabilities of large deviations for sums of independent random variables. Theor. Probab. Appl. 10 287–298.
• [19] Resnick, S.I. (2008). Extreme Values, Regular Variation and Point Processes. New York: Springer.
• [20] Rosenthal, H.P. (1970). On the subspaces of Lp spanned by sequences of independent random variables. Israel J. Math. 8 273–303.
• [21] Rvačeva, E.L. (1962). On domains of attraction of multi-dimensional distributions. In Select. Transl. Math. Statist. Probab. 2 183–205. Providence, RI: Amer. Math. Soc.
• [22] Samorodnitsky, G. and Taqqu, M. (1994). Stable Non–Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman Hall.
• [23] Stoev, S. (2008). On the ergodicity and mixing of max-stable processes. Stochastic Process. Appl. 118 1679–1705.