Math Problems Vault

Problem 12.1

If the independent r.v.s $X_{j}, j \geq 1$ , are distributed as $U(-j, j)$ , then show that the Lindeberg condition (see relation (12.24)) holds, so that $\mathcal{L}\left(\frac{S_{n}}{S_{n}}\right) \underset {n \rightarrow \infty }{\Rightarrow } N(0,1)$ , where $S_{n} = \sum_{j = 1}^{n} X_{j}$ and $s_{n}^{2} = \sum_{j = 1}^{n} \sigma_{j}^{2}, \sigma_{j}^{2} = \operatorname {Var}\left[X_{j}\right]$ .

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Problem 12.2

If the independent r.v.s $X_{j}, j \geq 2$ , are distributed as follows:

\begin{aligned} \mathbb {P}\left(X_{j} = -j^{\alpha }\right) = \mathbb {P}\left(X_{j} = j^{\alpha }\right) = \frac{1}{j^{\beta }}, \quad \mathbb {P}\left(X_{j} = 0\right) \\ & = 1-\frac{2}{j^{\beta }} \quad (\alpha , \beta > 0) \end{aligned}

show that the restriction $\beta < 1$ ensures that the Lindeberg condition (relation (12.24)) holds.

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Problem 12.3

Let the r.v.s $X_{j}, j \geq 1$ , be distributed as follows:

\begin{aligned} \mathbb {P}\left(X_{j} = \pm j^{\alpha }\right) = \frac{1}{6 j^{2(\alpha -1)}}, \quad \mathbb {P}\left(X_{j} = 0\right) = 1-\frac{1}{3 j^{2(\alpha -1)}}, \quad \alpha > 1 \end{aligned}

Show that the Lindeberg condition (relation (12.24)) holds, if and only if $\alpha < \frac{3}{2}$ . Conclude that $\mathcal{L}\left(\frac{S_{n}}{s_{n}}\right) \underset {n \rightarrow \infty }{\Rightarrow } N(0,1)$ , where $S_{n} = \sum_{j = 1}^{n} X_{j}$ and $s_{n}^{2} = \sum_{j = 1}^{n} \sigma_{j}^{2}$ , $\sigma_{j}^{2} = \operatorname {Var}\left[X_{j}\right]$ .

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Problem 12.4

For $j = 1,2, \ldots , n$ , let $X_{j}$ be independent r.v.s defined as follows:

X_{j} = \left\{ \begin{array}{l} \pm j^{2} \text{ with probability } 1 / 12 j^{2} \text{ each } \\ \pm j \text{ with probability } 1 / 12 \text{ each } \\ 0 \quad \text{ with probability } 1-1 / 6-1 / 6 j^{2} \end{array}\right.

Then show that the Lindeberg condition (condition (12.24)) does not hold.

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Problem 12.5

Let $X_{n}, n \geq 1$ , be independent r.v.s such that $\left|X_{n}\right| \leq M_{n}$ a.s. with $M_{n} = o\left(s_{n}\right)$ where $s_{n}^{2} = \sum_{j = 1}^{n} \sigma_{j}^{2} \rightarrow \infty$ and $\sigma_{j}^{2} = \operatorname {Var}\left[X_{j}\right]$ . Set $S_{n} = \sum_{j = 1}^{n} X_{j}$ and show that

\mathcal{L}\left(\frac{S_{n}-\mathbb {E}\left[S_{n}\right]}{s_{n}}\right) \underset {n \rightarrow \infty }{\Rightarrow } N(0,1)

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Problem 12.6

Let $X_{n j}, j = 1, \ldots , k_{n} \rightarrow \infty$ be row-wise independent r.v.s with $\mathbb {E}\left[X_{n j}\right] = 0$ , $\sigma^{2}\left(X_{n j}\right) = \sigma_{n j}^{2} < \infty , s_{n}^{2} = \sum_{j = 1}^{k_{n}} \sigma_{n j}^{2} = 1$ , and $\max_{1 \leq j \leq k_{n}} \sigma_{n j}^{2} \underset {n \rightarrow \infty }{\rightarrow } 0$ . Then, with $S_{n} = \sum_{j = 1}^{k_{n}} X_{n j}$ , and under the assumption that $\mathcal{L}\left(S_{n}\right) \Rightarrow N(0,1)$ , show that $\sum_{j = 1}^{k_{n}} X_{n j}^{2} \xrightarrow [\rightarrow ]{P} 1$ .

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Problem 12.7

(i)(ii)

For a function $g: \mathbb {R} \rightarrow \mathbb {C}$ , set $\Delta g(u) = g(u+h)-g(u-h), h \in \mathbb {R}$ , and define $\Delta^{(n)} g(u)$ recursively. Then

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(i)

Show that

\Delta ^{(m)} g(u) = \sum _{r = 0}^{m}(-1)^{r}\binom {m}{r} g(u+(m-2 r) h) .

(ii)

From part (i), and by expanding $f(t)$ around 0 up to terms of order $2 n$ , obtain

\begin{aligned} \frac{\Delta ^{(2 n)} f(0)}{(2 h)^{2 n}} = & f^{(2 n)}(0)+\frac{o\left(h^{2 n}\right)}{h^{2 n}} \\ & \left(\text{ where } \frac{o(t)}{t} \rightarrow 0 \text{ as } t \rightarrow 0\right), \end{aligned}

so that $f^{(2 n)}(0) = \lim \frac{\Delta^{(2 n)} f(0)}{(2 h)^{2 n}}$ as $h \rightarrow 0$ .

Problem 12.8

For $m = 1,2, \ldots$ and $k$ integer, show that

\begin{align} & \sum _{r = 0}^{m}(-1)^{r}\binom {m}{r}(m-2 r)^{k} \\ & \quad = \begin{cases} 0 & \text{ for }(0 \leq ) k < m \\ 2^{m} \times m! & k = m \end{cases} \end{align}

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Problem 12.9

For $j = 1, \ldots , k_{n} \rightarrow \infty$ , let $X_{n j}$ be row-wise independent r.v.s with respective d.f.s $F_{n j}$ . Then show that, for every $\varepsilon > 0$ , the convergence

\mathbb {P}\left(\max _{1 \leq j \leq k_{n}}\left|X_{n j}\right| \geq \varepsilon \right) \underset {n \rightarrow \infty }{\rightarrow } 0

is equivalent to

\sum _{j = 1}^{k_{n}} \int _{(\left|x\right| \geq \varepsilon )} d F_{n j}(x) \underset {n \rightarrow \infty }{\rightarrow } 0

and implies the convergence

\max _{1 \leq j \leq k_{n}} \mathbb {P}\left(\left|X_{n j}\right| \geq \varepsilon \right) \underset {n \rightarrow \infty }{\rightarrow } 0

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Problem 12.10

Show that, for every complex number $z$ , it holds $\left|z^{2}\right| = \left|z\right|^{2}$ .

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Problem 12.11

In the proof of Proposition 2 (see discussion following relation (12.36)), Lemma 6 is used for $n = 1$ to ensure finiteness of $\mathbb {E}\left[X^{2}\right]$ , which, according to the lemma, follows from the existence and finiteness of $f^{(2)}(0)$ . Prove this fact directly; i.e., show that the existence and finiteness of $f^{(2)}(0)$ implies finiteness of $\mathbb {E}\left[X^{2}\right]$ .

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