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Solutions to problems (b) to (e) of paper 79, focusing on large deviations and queues. Topics covered include rate functions, good rate functions, large deviations principle, lindley recursion, queue size function, downstream queue size function, and effective bandwidth. The document assumes a solid understanding of the concepts presented in the paper and the gärtner-ellis theorem.
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Monday 2 June 2003 1.30 to 3.
Attempt THREE questions.
There are four questions in total. The questions carry equal weight.
You may find helpful the reference material at the end of the paper.
1 Let A 1 , A 2 ,... be normal random variables with mean μ and variance σ^2. Let B be an exponential random variable with mean 1/λ. Let C be a normal random variable with mean ν and variance ρ^2. Let all of these random variables be independent.
(a) State, without proof, a large deviations principle for L−^1 B.
(b) Find a large deviations principle for L−^1 (A 1 + · · · + AL). (c) Find a large deviations principle for L−^1 (B + A 1 + · · · + AL).
(d) Find a large deviations principle for L−^1 (C + A 1 + · · · + AL).
(e) Comment on your results. State clearly any general results to which you appeal.
(^2) (a) Define these terms: rate function, good rate function, large deviations principle.
Recall that a sequence of random variables (XL, L ∈ N) is said to be exponentially tight if for all α > 0 there exists a compact set Kα such that
lim sup L→∞
log P (XL 6 ∈ Kα) < −α.
The sequence (XL, L ∈ N) is said to satisfy a weak large deviations principle if the large deviations upper bound is required to hold only for compact sets.
Suppose that the sequence (XL, L ∈ N) is exponentially tight, and satisfies a weak large deviations principle with rate function I.
(b) Show that I is a good rate function. (c) Show that the large deviations upper bound holds for closed sets.
Conclude that (XL, L ∈ N) satisfies a large deviations principle with good rate function I.
Paper 79
Reference: G¨artner-Ellis theorem
A convex function Λ : Rd^ → R ∪ {∞} is essentially smooth if
(a) the interior of its effective domain is non-empty (b) Λ(·) is differentiable throughout the interior of its effective domain
(c) Λ(·) is steep, namely, |∇Λ(θn)| → ∞ whenever (θn) is a sequence in the interior of the effective domain converging to a point on the boundary of the effective domain.
Let (XL, L ∈ N) be a sequence of random vectors in Rd, and let
ΛL(θ) =
log E exp(Lθ · XL)
for θ ∈ Rd. Assume that for each θ the limit
Λ(θ) = lim L→∞
ΛL(θ)
exists in R ∪ {∞}. Assume further that 0 is in the interior of the effective domain of Λ, and that Λ is essentially smooth and lower-semicontinuous. Then (XL, L ∈ N) satisfies an LDP in Rd^ with good rate function
Λ∗(x) = sup θ∈Rt
θ · x − Λ(θ).
Paper 79
