An Outlook on Further Topics

Let $X_1, X_2, \ldots$ be independent, equidistributed random variables, and set $S_n = X_1 + \cdots + X_n$, $n \geq 1$. The sequence $\left\{ S_n, n \geq 0\right\}$ (where $S_0 = 0$) is called a random walk. Consider the following ``perturbed'' random walk. Let $\left\{ \varepsilon_n, n \geq 1\right\}$ be a sequence of random variables such that, for some fixed $A > 0$, we have $\mathbb {P}\left(\left|\varepsilon_n\right| \leq A\right) = 1$ for all $n$, and set

Suppose that $\mathbb {E}\left[X_1\right] = \mu$ exists. Show that the law of large numbers holds for the perturbed random walk $\left\{ T_n, n \geq 1\right\}$.

In a game of dice one wishes to use one of two dice $A$ and $B$. $A$ has two white and four red faces and $B$ has two red and four white faces. A coin is tossed in order to decide which die is to be used and that die is then used throughout. Let $\left\{ X_k, k \geq 1\right\}$ be a sequence of random variables defined as follows:

at the $k$th roll of the die. Show that the law of large numbers does not hold for the sequence $\left\{ X_k, k \geq 1\right\}$. Why is this the case?

Suppose that $X_1, X_2, \ldots$ are independent random variables such that $X_k \in \operatorname {Ber}\left(p_k\right)$, $k \geq 1$, and set $S_n = \sum_{k=1}^{n} X_k$, $m_n = \sum_{k=1}^{n} p_k$, and $s_n^2 = \sum_{k=1}^{n} p_k(1-p_k)$, $n \geq 1$. Show that if

then

Prove the following central limit theorem for a sum of independent (not identically distributed) random variables: Suppose that $X_1, X_2, \ldots$ are independent random variables such that $X_k \in U(-k,k)$, and set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Show that

and determine $\mu$ and $\sigma^2$.

Let $X_1, X_2, \ldots$ be independent, $U(0,1)$-distributed random variables. We say that there is a peak at $X_k$ if $X_{k-1}$ and $X_{k+1}$ are both smaller than $X_k$, $k \geq 2$. What is the probability of a peak at

Verify formula (2.1), i.e., that if $X, X_1, X_2, \ldots$ are i.i.d.symmetric stable random variables, then

Prove that the law of large numbers holds for symmetric, stable distributions with index $\alpha$, $1 < \alpha \leq 2$.

Let $0 < \alpha < 2$ and suppose that $X, X_1, X_2, \ldots$ are independent random variables with common (two-sided Pareto) density

Show that the distribution belongs to the domain of attraction of a symmetric stable distribution with index $\alpha$; in other words, that the sums $S_n = \sum_{k=1}^{n} X_k$, suitably normalized, converge in distribution to a symmetric stable distribution with index $\alpha$.

The same problem as the previous one, but for the density

where $c$ is an appropriate normalizing constant.

Show that the extremal distributions belong to their own domain of attraction. More precisely, let $X, X_1, X_2, \ldots$ be i.i.d.random variables, and set

Show that,

Let $Y_1, Y_2, \ldots$ be independent random variables with mean zero and finite variances $\operatorname {Var}\left[Y_k\right] = \sigma_k^2$. Set

Show that $X_1, X_2, \ldots$ is a martingale.

Let $Y_1, Y_2, \ldots$ be i.i.d.random variables with finite mean $\mu$, and finite variance $\sigma^2$, and let $S_n$, $n \geq 1$, denote their partial sums. Set

Show that $X_1, X_2, \ldots$ is a martingale.

Let $X(n)$ be the number of individuals in the $n$th generation of a branching process ($X(0) = 1$) with reproduction mean $m$ ($= \mathbb {E}\left[X(1)\right]$). Set

Show that $U_1, U_2, \ldots$ is a martingale.

Let $Y_1, Y_2, \ldots$ are i.i.d.random variables with a finite moment generating function $\psi$, set $S_n = \sum_{k=1}^{n} Y_k$, $n \geq 1$, with $S_0 = 0$, and