# 微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Random Indices

• 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|Random Indices

Let $\tau_{n}$ be an $\mathbb{N}$-valued random variable for every $n \in \mathbb{N}$. We are interested in the convergence of $\left(X_{\tau_{n}}\right){n \geq 1}$ for $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$-valued random variables $X{n}$ provided $\tau_{n} \rightarrow \infty$ in probability as $n \rightarrow \infty$, that is $\lim {n \rightarrow \infty} P\left(\tau{n} \geq C\right)=1$ for every $C \in(0, \infty)$.

We start with the simple independent setting where $\left(\tau_{n}\right){n \geq 1}$ and $\left(X{n}\right)_{n \geq 1}$ are independent. Here we observe that stable convergence is preserved by such a random time change with the same limit.

## 微积分网课代修|极限理论代写Limit Theory代考|The Empirical Measure Theorem and the δ-Method

The following result (see [7], Corollary 3.16, Theorem 4.7, [31]) allows us to pass from stable convergence to almost sure convergence and has the Komlós theorem as its point of departure.

Theorem $4.9$ (Empirical measure theorem) If $X_{n} \rightarrow K$ stably for $(\mathcal{X}, \mathcal{B}(\mathcal{X})$ )valued random variables $X_{n}$ and $K \in \mathcal{K}^{1}$, then there exists a subsequence $\left(X_{m}\right)$ of $\left(X_{n}\right)$ such that for every further subsequence $\left(X_{k}\right)$ of $\left(X_{m}\right)$, almost surely
$$\frac{1}{r} \sum_{k=1}^{r} \delta_{X_{k}(\omega)} \rightarrow K(\omega, \cdot) \quad \text { weakly }\left(\text { in } \mathcal{M}^{1}(\mathcal{X})\right) \text { as } r \rightarrow \infty$$
The above assertion simply means almost sure convergence of $\frac{1}{r} \sum_{k=1}^{r} \delta_{X_{k}}$ to $K$ when the Markov kernels are seen as $\left(\mathcal{M}^{1}(\mathcal{X}), \mathcal{B}\left(\mathcal{M}^{1}(\mathcal{X})\right)\right)$-valued random variables. Note that the exceptional null set may vary with the subsequence. In general, the assertion is not true for $\left(X_{n}\right)$ itself (see [7], Example 3.17). However, in the classical case of an independent and identically distributed sequence $\left(X_{n}\right)$ it is well known that $\left(X_{n} \rightarrow P^{X_{1}}\right.$ mixing and $)$ almost surely
$$\frac{1}{r} \sum_{n=1}^{r} \delta_{X_{n}}(\omega) \rightarrow P^{X_{1}} \quad \text { weakly as } r \rightarrow \infty$$

## 微积分网课代修|极限理论代写Limit Theory代考|The Empirical Measure Theorem

and the $\delta$-Method 以下结果（见[7]，推论3.16，定理4.7，[31]）允许我们从稳定收敛到几乎肯定的收 敛，并以Komlós定理为出发点。 定理 $4.9$ (实证测度定理) 如果 $X{n} \rightarrow K$ 稳定用于 $\left(\mathcal{X}, \mathcal{B}(\mathcal{X})\right.$ ) 值随机变量 $X_{n}$ 和 $K \in \mathcal{K}^{1}$ ，则存在子序列 $\left(X_{m}\right)$ 之 $\left(X_{n}\right)$ 使得对于每进一步的子序列 $\left(X_{k}\right)$ 之 $\left(X_{m}\right)$ ， 几乎可以肯定
$$\frac{1}{r} \sum_{k=1}^{r} \delta_{X_{k}(\omega)} \rightarrow K(\omega, \cdot) \quad \text { weakly }\left(\text { in } \mathcal{M}^{1}(\mathcal{X})\right) \text { as } r \rightarrow \infty$$

$$\frac{1}{r} \sum_{n=1}^{r} \delta_{X_{n}}(\omega) \rightarrow P^{X_{1}} \quad \text { weakly as } r \rightarrow \infty$$