# 微积分网课代修|极限理论代写Limit Theory代考|MATH6710 Counterexamples

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|极限理论代写Limit Theory代考|Counterexamples

This section will shed some light on the role of the conditions in Theorem 6.1. The first result shows that the row sums of a square integrable martingale difference array have weak limit points if the row sums of the conditional variances are bounded in probability.

Proposition 6.9 Let $\left(X_{n k}\right){1 \leq k \leq k{n}, n \in \mathbb{N}}$ be a square integrable martingale difference array adapted to an array $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}, n \in \mathbb{N}}$ of $\sigma$-fields. If the sequence $\left(\sum_{k=1}^{k_{n}} E\left(X_{n k}^{2} \mid \mathcal{F}{n, k-1}\right)\right){n \in \mathbb{N}}$ is bounded in probability, then the sequence $\left(\sum_{k=1}^{k_{n}} X_{n k}\right)_{n \in \mathbb{N}}$ is also bounded in probability.

Note that for sequences of real (or $\mathbb{R}^{d}$-valued) random variables boundedness in probability is the same as tightness.

## 微积分网课代修|极限理论代写Limit Theory代考|Further Sufficient Conditions

The conditions (N) and (CLB) in Theorem $6.1$ may be replaced by several other sets of sufficient conditions. Some of these will be introduced and discussed in this section, which is partly based on [34]. We always consider an array $\left(X_{n k}\right){1 \leq k \leq k{n}, n \in \mathbb{N}}$ of random variables and an array $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}, n \in \mathbb{N}}$ of sub- $\sigma$-fields of $\mathcal{F}$ for some basic probability space $(\Omega, \mathcal{F}, P)$. The $\sigma$-fields $\mathcal{G}{n k}$ and $\mathcal{G}$ are defined as in Theorem 6.1. For a square integrable array $\left(X{n k}\right){1 \leq k \leq k{n}, n \in \mathbb{N}}$ we introduce the condition

$\left(\mathrm{M}{2}\right) \quad E\left(\max {1 \leq k \leq k_{n}} X_{n k}^{2}\right) \rightarrow 0$ as $n \rightarrow \infty$
whereas the conditions
$\left(\mathrm{M}{1}\right) \quad E\left(\max {1 \leq k \leq k_{n}}\left|X_{n k}\right|\right) \rightarrow 0$ as $n \rightarrow \infty$
and
$\left(\mathrm{CLB}{1}\right) \quad \sum{k=1}^{k_{n}} E\left(\left|X_{n k}\right| 1_{\left{\left|X_{n k}\right| \geq \varepsilon\right}} \mid \mathcal{F}{n, k-1}\right) \rightarrow 0 \quad$ in probability as $n \rightarrow \infty$ for every $\varepsilon>0$ can be imposed on any array $\left(X{n k}\right){1 \leq k \leq k{n}, n \in \mathbb{N}}$ of integrable random variables.

## 微积分网课代修|极限理论代写Limit Theory代考|Counterexamples

$(\mathcal{F} n k) 0 \leq k \leq k_{n}, n \in \mathbb{N}$ 的 $\sigma$-字段。如果序列
$\left(\sum_{k=1}^{k_{n}} E\left(X_{n k}^{2} \mid \mathcal{F} n, k-1\right)\right) n \in \mathbb{N}$ 是概率有界的，那么序列
$\left(\sum_{k=1}^{k_{n}} X_{n k}\right){n \in \mathbb{N}}$ 也是有界的概率。 请注意，对于实数序列 (或 $\mathbb{R}^{d}$-valued) 随机变量的概率有界性与紧密性相同。

## 微积分网课代修|极限理论代写Limit Theory代考|Further Sufficient Conditions

$(\Omega, \mathcal{F}, P)$. 这 $\sigma$-字段 $\mathcal{G} n k$ 和 $\mathcal{G}$ 定义如定理 6.1。对于正方形可积数组
$(X n k) 1 \leq k \leq k n, n \in \mathbb{N}$ 我们引入条件
(M2) $E\left(\max 1 \leq k \leq k_{n} X_{n k}^{2}\right) \rightarrow 0$ 作为 $n \rightarrow \infty$

(M1) $E\left(\max 1 \leq k \leq k_{n}\left|X_{n k}\right|\right) \rightarrow 0$ 作为 $n \rightarrow \infty$