Large deviations of inverse processes with nonlinear scalings. (English) Zbl 0939.60011

Duffield, N. G.; Whitt, W.

Ann. Appl. Probab. 8, No. 4, 995-1026 (1998).

For a right-continuous, nonnegative, nondecreasing stochastic process \((Z_t)_{t\geq 0}\) with \(\lim_{t\to\infty }Z_t=\infty \) we define the inverse process \(T_z=\inf\{t\geq 0, Z_t\geq z\}\), \(z\geq 0\), which is a left-continuous, nonnegative, nondecreasing process with \(T_0=0\) and \(\lim_{t\to\infty }T_t=\infty \). Then, under regularity conditions, \(u(Z_t)/w(t)\) satisfies the large deviation principle (LDP) with a scaling function \(v(t)\) and a rate function \(I\) if and only if \(w(T_z)/u(z)\) satisfies the LDP with the scaling function \(v\circ w^{-1}\circ u(z)\) and the rate function \(J(x)=f(x)I(1/x)\) where \(f(x)=x^{i(f)}\) with \(i(f)\) being the index of the regularly varying function \(v\circ w^{-1}\). An application to queueing models as well as examples concerning fractional Gaussian noise, time-transformed stationary processes, compound processes and renewal theory are also given.

Reviewer: Ilya Pavlyukevitch (Berlin)

MSC:

60F10	Large deviations
60K25	Queueing theory (aspects of probability theory)
60G18	Self-similar stochastic processes

Keywords:

large deviations; renewal theory; queueing theory; regularly varying functions; inverse processes; counting processes; fractional Gaussian noise

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