Let X1, …, Xn be independent Poisson trials such that Pr(Xi=1)= p. Let X=∑i=1^n Xi, so that 𝐄[X]

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Let X1, …, Xn be independent Poisson trials such that...

Let $X_{1}, \ldots, X_{n}$ be independent Poisson trials such that $\operatorname{Pr}\left(X_{i}=1\right)=$ $p$. Let $X=\sum_{i=1}^{n} X_{i}$, so that $\mathbf{E}[X]=p n$. Let $$ F(x, p)=x \ln (x / p)+(1-x) \ln ((1-x) /(1-p)) . $$ (a) Show that, for $1 \geq x>p$, $$ \operatorname{Pr}(X \geq x n) \leq \mathrm{e}^{-n F(x, p)} $$ (b) Show that, when $0<x, p<1$, we have $F(x, p)-2(x-p)^{2} \geq 0$. (Hint: Take the second derivative of $F(x, p)-2(x-p)^{2}$ with respect to $x$.) (c) Using parts (a) and (b), argue that $$ \operatorname{Pr}(X \geq(p+\varepsilon) n) \leq \mathrm{e}^{-2 n \varepsilon^{2}} $$ (d) Use symmetry to argue that $$ \operatorname{Pr}(X \leq(p-\varepsilon) n) \leq \mathrm{e}^{-2 n \varepsilon^{2}}, $$ and conclude that $$ \operatorname{Pr}(|X-p n| \geq \varepsilon n) \leq 2 \mathrm{e}^{-2 n \varepsilon^{2}} $$

Let $X_{1}, \ldots, X_{n}$ be independent Poisson trials such that $\operatorname{Pr}\left(X_{i}=1\right)=$ $p$. Let $X=\sum_{i=1}^{n} X_{i}$, so that $\mathbf{E}[X]=p n$. Let
$$
F(x, p)=x \ln (x / p)+(1-x) \ln ((1-x) /(1-p)) .
$$
(a) Show that, for $1 \geq x>p$,
$$
\operatorname{Pr}(X \geq x n) \leq \mathrm{e}^{-n F(x, p)}
$$
(b) Show that, when $0<x, p<1$, we have $F(x, p)-2(x-p)^{2} \geq 0$. (Hint: Take the second derivative of $F(x, p)-2(x-p)^{2}$ with respect to $x$.)
(c) Using parts (a) and (b), argue that
$$
\operatorname{Pr}(X \geq(p+\varepsilon) n) \leq \mathrm{e}^{-2 n \varepsilon^{2}}
$$
(d) Use symmetry to argue that
$$
\operatorname{Pr}(X \leq(p-\varepsilon) n) \leq \mathrm{e}^{-2 n \varepsilon^{2}},
$$
and conclude that
$$
\operatorname{Pr}(|X-p n| \geq \varepsilon n) \leq 2 \mathrm{e}^{-2 n \varepsilon^{2}}
$$

Key Concepts

Large Deviations

Large deviation theory studies the asymptotic behavior of remote tails of sequences of probability distributions. It provides a framework in which the probability of rare events, such as substantial deviations of sums of independent random variables from their mean, decays exponentially with the number of observations. The function F(x, p) serves as the rate function in this framework, quantifying the exponential rate at which these probabilities shrink.

Concentration Inequalities

Concentration inequalities provide bounds indicating that a random variable is highly likely to be close to a certain value, such as its mean. They ensure that the probability of significant deviation decays rapidly, usually exponentially, with the number of observations. The problem employs concentration inequalities to show that the probability of the sum of independent Bernoulli trials deviating from its mean is at most 2e^(-2n?^2).

Chernoff Bound

The Chernoff bound is an exponential tail bound used to estimate the probability that the sum of independent random variables deviates significantly from its expected value. In this context, it is applied to bound the probabilities of a binomial random variable exceeding or falling below certain thresholds by using the moment generating function and an optimization over a parameter, leading to exponential decay bounds in the form of exp(-n F(x, p)).

Kullback-Leibler Divergence

Kullback-Leibler divergence, also known as relative entropy, quantifies the difference between two probability distributions. In this problem, the function F(x, p) takes the form of the divergence between two Bernoulli distributions with parameters x and p. This divergence measures how 'surprised' one is to see a proportion x when expecting p, thereby naturally appearing in bounds for the probabilities of large deviations.

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