On the asymptotic behaviour of probabilities of large deviations for the negative polynomial distribution
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A.N. Timashov
Abstract
We consider the polynomial scheme of trials with outcomes E0, E1, ... , EN and the corresponding probabilities p0, p1 , ... , pN. We assume that the trials are performed until the rth occurrence of the outcome E0, r = 1, 2, ... If ηj(r) is the number of occurrences of the outcome Ej at the stopping time, j = 1, ... , N, and η(r) = (η1(r), ... , ηN(r)), then the vector η(r) has the negative polynomial distribution. Under the assumptions that N ∈ ℕ and the positive probabilities p0, p1, ... , pN are fixed, that r → ∞ and k1, ... , kN → ∞ so that the parameters .βj = kj/r satisfy the inequalities βj ≥ ε, where ε is a positive constant, j = 1, ... , N, and under some additional constraints, we give asymptotic estimates of the probabilities of large deviations
P{ηj(r) ≤ kj, j = 1, ... , N}, P{ηj(r) ≥ kj, j = 1, ... , N}.
In order to derive these asymptotic estimates, we use the multidimensional saddle-point method in the form suggested by Good.
© 2016 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Editorial
- Final probabilities for modified branching processes
- Non-interference models and subliminal channels
- Monic polylinear shift registers and their periods
- Limit distributions of the maximum size of a tree in a random recursive forest
- On the asymptotic behaviour of probabilities of large deviations for the negative polynomial distribution
- On the functional complexity of a two-dimensional interval search problem
- On analytical conjugacy of Ton chard polynomials and the polynomials quasi-orthogonal to them
- Forthcoming Papers
- Contents
Articles in the same Issue
- Editorial
- Final probabilities for modified branching processes
- Non-interference models and subliminal channels
- Monic polylinear shift registers and their periods
- Limit distributions of the maximum size of a tree in a random recursive forest
- On the asymptotic behaviour of probabilities of large deviations for the negative polynomial distribution
- On the functional complexity of a two-dimensional interval search problem
- On analytical conjugacy of Ton chard polynomials and the polynomials quasi-orthogonal to them
- Forthcoming Papers
- Contents
