Convergence

Let $X_1, X_2, \ldots$ be $U(0,1)$-distributed random variables. Show that

Let $\left\{ X_n, n \geq 1\right\}$ be a sequence of i.i.d. random variables with density

Set $Y_n = \min \left\{ X_1, X_2, \ldots , X_n\right\}$. Show that

Let $X_k$, $k \geq 1$, be i.i.d. random variables with finite variance $\sigma^2$, and let, for $n \geq 1$,

denote the arithmetic mean and sample variance, respectively. It is well known(?) that $s_n^2$ is an unbiased estimator of $\sigma^2$, that is, that $\mathbb {E}\left[s_n^2\right] = \sigma^2$.

Let $(X_k, Y_k)$, $1 \leq k \leq n$, be a sample from a two-dimensional distribution with mean vector and covariance matrix

respectively, and let

denote the arithmetic means and sample variances of the respective components. The empirical correlation coefficient is defined as

Prove that

Let $X_1, X_2, \ldots$ be independent, $C(0,1)$-distributed random variables. Determine the limit distribution of

as $n \rightarrow \infty$.

Suppose that $X_1, X_2, \ldots$ are independent, $\text{Pa}(1,2)$-distributed random variables, and set $Y_n = \max \left\{ X_1, X_2, \ldots , X_n\right\}$.

Let $X_1, X_2, \ldots$ be i.i.d. random variables, and let

Suppose that $X_n \in \text{Ge}(\lambda /(n+\lambda ))$, $n = 1, 2, \ldots$, where $\lambda$ is a positive constant. Show that $X_n/n$ converges in distribution to an exponential distribution as $n \rightarrow \infty$, and determine the parameter of the limit distribution.

Let $X_1, X_2, \ldots$ be a sequence of random variables such that

Determine the limit distribution of $X_n$ as $n \rightarrow \infty$.

Let $X_n \in \text{Bin}(n, p_n)$.

Let $X_n \in \text{Bin}(n^2, m/n)$, $m > 0$. Show that

Let $X_{n1}, X_{n2}, \ldots , X_{nn}$ be independent random variables with a common distribution given as follows:

where $k = 1, 2, \ldots , n$ and $n = 2, 3, \ldots$. Set $S_n = X_{n1} + X_{n2} + \cdots + X_{nn}$, $n \geq 2$. Show that

Let $X_1, X_2, \ldots$ be independent, equidistributed random variables with characteristic function

Set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Show that $S_n/n^2$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Let $X_{n1}, X_{n2}, \ldots , X_{nn}$ be independent random variables, with a common distribution given, and set

and $S_n = X_{n1} + X_{n2} + \cdots + X_{nn}$ for $k = 1, 2, \ldots , n$ and $n \geq 2$. Show that

and determine $\lambda$.

Let $X$ and $Y$ be random variables such that

Let $X_1, X_2, \ldots$ be independent, $L(a)$-distributed random variables, and let $N \in \text{Po}(m)$ be independent of $X_1, X_2, \ldots$. Determine the limit distribution of $S_N = X_1 + X_2 + \cdots + X_N$ (where $S_0 = 0$) as $m \rightarrow \infty$ and $a \rightarrow 0$ in such a way that $m \cdot a^2 \rightarrow 1$.

Let $N, X_1, X_2, \ldots$ be independent random variables such that $N \in \text{Po}(\lambda )$ and $X_k \in \text{Po}(\mu )$, $k = 1, 2, \ldots$. Determine the limit distribution of $Y = X_1 + X_2 + \cdots + X_N$ as $\lambda \rightarrow \infty$ and $\mu \rightarrow 0$ such that $\lambda \cdot \mu \rightarrow \gamma > 0$. (The sum is zero for $N = 0$.)

Let $X_1, X_2, \ldots$ be independent $\text{Po}(m)$-distributed random variables, suppose that $N \in \text{Ge}(p)$ is independent of $X_1, X_2, \ldots$, and set $S_N = X_1 + X_2 + \cdots + X_N$ (and $S_0 = 0$ for $N = 0$). Let $m \rightarrow 0$ and $p \rightarrow 0$ in such a way that $p/m \rightarrow \alpha$, where $\alpha$ is a given positive number. Show that $S_N$ converges in distribution, and determine the limit distribution.

Suppose that the random variables $N_n, X_1, X_2, \ldots$ are independent, that $N_n \in \text{Ge}(p_n)$, $0 < p_n < 1$, and that $X_1, X_2, \ldots$ are equidistributed with finite mean $\mu$. Show that if $p_n \rightarrow 0$ as $n \rightarrow \infty$ then $p_n(X_1 + X_2 + \cdots + X_{N_n})$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Suppose that $X_1, X_2, \ldots$ are i.i.d. symmetric random variables with finite variance $\sigma^2$, let $N_p \in \text{Fs}(p)$ be independent of $X_1, X_2, \ldots$, and set $Y_p = \sum_{k=1}^{N_p} X_k$. Show that

and determine $a$.

Let $X_1, X_2, \ldots$ be independent, $U(0,1)$-distributed random variables, and let $N_m \in \text{Po}(m)$ be independent of $X_1, X_2, \ldots$. Set

($V_m = 0$ when $N_m = 0$). Determine

Let $X_{1n}, X_{2n}, \ldots , X_{nn}$ be independent random variables such that $X_{kn} \in \text{Be}(p_{k,n})$, $k = 1, 2, \ldots , n$, $n \geq 1$. Suppose, further, that $\sum_{k=1}^{n} p_{k,n} \rightarrow \lambda < \infty$ and that $\max_{1 \leq k \leq n} p_{k,n} \rightarrow 0$ as $n \rightarrow \infty$. Show that $\sum_{k=1}^{n} X_{kn}$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Show that

by applying the central limit theorem to suitably chosen, independent, Poisson-distributed random variables.

Let $X_1, X_2, \ldots$ be independent, $U(-1,1)$-distributed random variables.

Let $X_n \in \Gamma (n,1)$, and set

Show that $Y_n \xrightarrow {d} N(0,1)$ as $n \rightarrow \infty$.

Let $X_1, X_2, \ldots$ be positive, i.i.d. random variables with mean $\mu$ and finite variance $\sigma^2$, and set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Show that

and determine $b^2$.

Let $X_1, X_2, \ldots$ be i.i.d. random variables with finite mean $\mu \neq 0$, and set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$.

Suppose that $X_1, X_2, \ldots$ are i.i.d. random variables with mean 0 and variance 1. Show that

Let $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ be two (not necessarily independent) sequences of random variables and suppose that $X_n \xrightarrow {d} X$ and that $\mathbb {P}\left(X_n \neq Y_n\right) \rightarrow 0$ as $n \rightarrow \infty$. Prove that (also)

Let $\left\{ Y_k, k \geq 1\right\}$ be independent, $U(-1,1)$-distributed random variables, and set

Show that $X_n \xrightarrow {d} N(0,1/3)$ as $n \rightarrow \infty$.

Let $X_1, X_2, \ldots$ be independent, $U(-a,a)$-distributed random variables ($a > 0$). Set

Show that $S_n Z_n / V_n$, suitably normalized, converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Let $X_1, X_2, \ldots$ be independent, $U(0,1)$-distributed random variables, and set

Determine the limit distribution of $nV_n/Z_n$ as $n \rightarrow \infty$.

Let $X_1, X_2, \ldots$ be independent random variables such that $X_k \in \text{Exp}(k!)$, $k = 1, 2, \ldots$, and set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Show that

Let $X_1, X_2, \ldots$ be i.i.d. random variables with expectation 1 and finite variance $\sigma^2$, and set $S_n = X_1 + X_2 + \cdots + X_n$, for $n \geq 1$. Show that

and determine the constant $b^2$.

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and positive, finite variance $\sigma^2$, and set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Finally, suppose that $g$ is twice continuously differentiable, and that $g'(\mu ) \neq 0$. Show that

and determine $b^2$.

Let $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ be independent sequences of independent random variables. Suppose that there exist sequences $\left\{ a_n, n \geq 1\right\}$ of real numbers and $\left\{ b_n, n \geq 1\right\}$ of positive real numbers tending to infinity such that

where $Z_1$ and $Z_2$ are independent random variables. Show that

Suppose that $\left\{ U_t, t \geq 0\right\}$ and $\left\{ V_t, t \geq 0\right\}$ are families of random variables, such that

for some finite constant $a$, and random variable $V$. Prove that

as $t \rightarrow \infty$.

Let $X_1, X_2, \ldots$ be independent random variables such that, for some fixed positive integer $m$, $X_1, \ldots , X_m$ are equidistributed with mean $\mu_1$ and variance $\sigma_1^2$, and $X_{m+1}, X_{m+2}, \ldots$ are equidistributed with mean $\mu_2$ and variance $\sigma_2^2$. Set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Show that the central limit theorem (still) holds.

Let $X_1, X_2, \ldots$ be $U(-1,1)$-distributed random variables, and set

Let $Z \in U(0,1)$ be independent of $Y_1, Y_2, \ldots$, where

and set

Let $X_1, X_2, \ldots$ be identically distributed random variables converging in distribution to the random variable $X$, let $\left\{ a_n, n \geq 1\right\}$ and $\left\{ b_n, n \geq 1\right\}$ be sequences of positive reals $\nearrow +\infty$, and set

The following example shows that a sequence of continuous random variables may converge in distribution without the sequence of densities being convergent. Namely, let $X_n$ have a distribution function given by

Show that $X_n \xrightarrow {d} U(0,1)$ as $n \rightarrow \infty$, but that $f_n(x)$ does not converge to the density of the $U(0,1)$-distribution.

Let $Y_1, Y_2, \ldots$ be a sequence of random variables such that

and let $\left\{ N_n, n \geq 1\right\}$ be a sequence of nonnegative, integer-valued random variables such that

Finally, suppose that the sequences $\left\{ N_n, n \geq 1\right\}$ and $Y_1, Y_2, \ldots$ are independent of each other. Prove (for example, with the aid of characteristic functions) that

Let $X_1, X_2, \ldots$ and $X$ be normal random variables. Show that

Suppose that $X_n \in \text{Exp}(a_n)$, $n \geq 1$. Show that

Prove that if $\left\{ X_n, n \geq 1\right\}$ are Poissonian random variables such that $X_n$ converges to $X$ in square mean, then $X$ must be Poissonian too.

Suppose that $\left\{ Z_k, k \geq 1\right\}$ is a sequence of branching processes---all of them starting with one single individual at time 0. Suppose, furthermore, that $Z_k \xrightarrow {d} Z$ as $k \rightarrow \infty$, where $Z$ is some nonnegative integer-valued random variable. Show that the corresponding sequence $\left\{ \eta_k, k \geq 1\right\}$ of extinction probabilities converges as $k \rightarrow \infty$.

Let $X_1, X_2, \ldots$ be independent random variables such that $X_k \in \text{Po}(k)$, $k = 1, 2, \ldots$, and set $Z_n = \frac{1}{n}\left(\sum_{k=1}^{n} X_k - n^2/2\right)$. Show that $Z_n$ converges in distribution as $n \rightarrow \infty$, and determine the limit distribution.

Suppose that $N \in \text{Po}(\lambda )$ independent observations of a random variable, $X$, with mean 0 and variance 1, are performed. Moreover, assume that $N$ is independent of $X_1, X_2, \ldots$. Show that

The purpose of this problem is to show that one can obtain a (kind of) central limit theorem even if the summands have infinite variance (if the variance does not exist). A short introduction to the general topic of possible limit theorems for normalized sums without finite variance is given in Section 7.3.

Let $X_1, X_2, \ldots$ be independent random variables with the following symmetric Pareto distribution:

Set $S_n = \sum_{k=1}^{n} X_k$, $n \geq 1$. Show via the following steps that

(Note that we do not normalize by $\sqrt{n}$ as in the standard case.)

Fix $n$ and consider, for $k = 1, 2, \ldots , n$, the truncated random variables

and

and note that $Y_{nk} + Z_{nk} = X_k$. Further, set $S_n' = \sum_{k=1}^{n} Y_{nk}$ and $S_n'' = \sum_{k=1}^{n} Z_{nk}$.