## Heavy-tailed random walks, buffered queues and hidden large deviations.(English)Zbl 1477.60052

Let $$(Z_i)_{i\in \mathbb{N}}$$ be an independent and identically distributed sequence with $$\alpha$$-regularly varying tails ($$\alpha > 0$$) such that an appropriate tail-balance condition holds. When $$Z_1$$ has finite mean assume additionally that the $$Z_i$$ are centered. Consider the random walk $$(S_n)_{n\in \mathbb{N}}$$ defined by $$S_n = \sum_{k=1}^n Z_k$$ and its càdlàg embedding $$X^{(n)}$$ in the space $$\mathbb{D}$$ of càdlàg functions on $$[0,1]$$, defined by $$X^{(n)}(t) = S_{\lfloor nt \rfloor}$$. Let $$(\lambda_n)_{n\in \mathbb{N}}$$ be a $$\rho$$-regularly varying sequence of real numbers, where $$\rho>1/2$$ when $$Z_1$$ has finite variance and $$\rho > 1/\alpha$$ when $$Z_1$$ has infinite variance. For each $$j\in \mathbb{N}$$, denote by $$\mathbb{D}_{\leq (j-1)}$$ the set of càdlàg step functions on $$[0,1]$$ with at most $$j-1$$ jumps, and let $$\gamma_n^{(j)} := [n P(|Z_1|>\lambda_n)]^{-j}$$. The authors then show that $$\gamma_n^{(j)} P(X^{(n)}/\lambda_n \in \cdot)$$ converges to a non-degenerate measure $$\mu_j$$ as $$n\to\infty$$, where the measure $$\mu_j$$ is given explicitly. Here, the convergence concept is the so called $$\mathbb{M}_{\mathbb{O}}$$-convergence in the space $$\mathbb{M}_{\mathbb{D}\setminus \mathbb{D}_{\leq (j-1)}}$$ of all Borel measures on $$\mathbb{D} \setminus \mathbb{D}_{\leq (j-1)}$$ that are bounded away from $$\mathbb{D}_{\leq (j-1)}$$. The used $$\mathbb{M}_{\mathbb{O}}$$-convergence concept, which is discussed in detail in [F. Lindskog et al., Probab. Surv. 11, 270–314 (2014; Zbl 1317.60007)], is related to the usual concept of weak convergence of measures. The result described above is a hidden large deviation type limit result. For $$j=1$$ it corresponds to a large deviation result on the space of càdlàg functions established by H. Hult et al. [Ann. Appl. Probab. 15, No. 4, 2651–2680 (2005; Zbl 1166.60309)]. The key point of the proved theorem is that it also works for any $$j\in \mathbb{N}$$, hence allowing for hidden large deviations of higher orders when $$j\geq 2$$ is considered. The results are then applied in the context of queuing processes with heavy-tailed service times. They are also used to provide probability estimates of rare events governed by more than one large jump in the case of random walks with infinite mean.

### MSC:

 60F10 Large deviations 60F17 Functional limit theorems; invariance principles 60G70 Extreme value theory; extremal stochastic processes 60G50 Sums of independent random variables; random walks

### Citations:

Zbl 1317.60007; Zbl 1166.60309
Full Text:

### References:

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