Math Problems Vault

Problem 5.11.1

A fair coin is tossed $n$ times, showing heads $H_{n}$ times and tails $T_{n}$ times. Let $S_{n} = H_{n}-T_{n}$ . Show that

\mathbb {P}\left(S_{n} > a n\right)^{1 / n} \rightarrow \frac{1}{\sqrt{(1+a)^{1+a}(1-a)^{1-a}}} \quad \text{ if } 0 < a < 1

What happens if $a \geq 1$ ?

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Problem 5.11.2

Show that

T_{n}^{1 / n} \rightarrow \frac{4}{\sqrt{(1+a)^{1+a}(1-a)^{1-a}}}

as $n \rightarrow \infty$ , where $0 < a < 1$ and

T_{n} = \sum _{\substack {k: \\ \left|k-\frac{1}{2} n\right| > \frac{1}{2} a n}}\binom {n}{k}

Find the asymptotic behaviour of $T_{n}^{1 / n}$ where

T_{n} = \sum _{\substack {k \cdot \\ k > n(1+a)}} \frac{n^{k}}{k!}, \quad \text{ where } a > 0

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Problem 5.11.3

Show that the moment generating function of $X$ is finite in a neighbourhood of the origin if and only if $X$ has exponentially decaying tails, in the sense that there exist positive constants $\lambda$ and $\mu$ such that $\mathbb {P}\left(\left|X\right| \geq a\right) \leq \mu e^{-\lambda a}$ for $a > 0$ . [Seen in the light of this observation, the condition of the large deviation theorem (5.11.4) is very natural].

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Problem 5.11.4

Let $X_{1}, X_{2}, \ldots$ be independent random variables having the Cauchy distribution, and let $S_{n} = X_{1}+X_{2}+\cdots +X_{n}$ . Find $\mathbb {P}\left(S_{n} > a n\right)$ .

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Problem 5.11.5

Chernoff inequality for Bernoulli trials. Let $S = \sum_{r = 1}^{n} X_{r}$ be a sum of independent Bernoulli random variables $X_{r}$ taking values in $\left\{ 0,1\right\}$ , where $\mathbb {E}\left[X_{r}\right] = p_{r}$ and $\mathbb {E}\left[S\right] = \mu > 0$ . Show that

\mathbb {P}\left(S > (1+\epsilon ) \mu \right) \leq \exp \left\{ -\mu [(1+\epsilon ) \log (1+\epsilon )-\epsilon ]\right\} , \quad \epsilon > 0

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