# 数学代写|概率论代写Probability theory代考|Strong Law of Large Numbers

## 数学代写|概率论代写Probability theory代考|Strong Law of Large Numbers

Applications of martingales are numerous. One application is to prove the Strong Law of Large Numbers (SLLN). This theorem says that if $Z_1, Z_2, \ldots$ is a sequence of integrable independent and identically distributed r.r.v’s with mean 0 , then $n^{-1}\left(Z_1+\cdots+Z_n\right) \rightarrow 0$ a.u. Historically, the first proof of this theorem in its generality, due to Kolmogorov, is constructive, complete with rates of convergence. See, for example, theorem 5.4 .2 of [Chung 1968]. Subsequently, remarkable proofs are also given in terms of a.u. martingale convergence via Doob’s upcrossing inequality. See, for example, theorem 9.4.1 of [Chung 1968]. As observed earlier, the theorem that deduces a.u. convergence from upcrossing inequalities actually implies the principle of infinite search, and cannot be made constructive. In this section, we present a constructive proof by a simple application of our Theorem 8.3.5.

First we discuss the Weak Law of Large Numbers, with a well-known proof by characteristic functions.

Theorem 8.4.1. Weak Law of Large Numbers. Suppose $Z_1, Z_2, \ldots$ is a sequence of integrable, independent, and identically distributed r.r.v’s with mean 0 , on some probability space $(\Omega, L, E)$. Let $\eta$ be a simple modulus of integrability of $Z_1$, in the sense of Definition 4.7.2. For each $m \geq 1$, let $S_m \equiv m^{-1}\left(Z_1+\cdots+Z_m\right)$. Then
$$E\left|S_m\right| \rightarrow 0$$
as $m \rightarrow \infty$. More precisely, there exists a sequence $\left(q_m\right){m=1,2, \ldots}$ of positive integers, which depends only on $\eta$ and is such that $$b_m \equiv \sup {k \geq q(m)} E\left|S_k\right| \leq 2^{-m}$$
for each $m \geq 1$.
Proof. 1. First note that, by Proposition 4.7.1, for each $k \geq 1$, the r.r.v. $Z_k$ has a modulus of integrability $\delta$ defined by
$$\delta(\varepsilon) \equiv \frac{\varepsilon}{2} / \eta\left(\frac{\varepsilon}{2}\right)$$
for each $\varepsilon>0$.

1. By hypothesis, the independent r.r.v.’s $Z_1, Z_2, \ldots$ have a common distribution $J$ on $R$. Hence they share a common characteristic function $\psi$. Therefore, for each $k \geq 1$, the characteristic function of the r.r.v. $S_k$ is given by
$$\psi_k \equiv \psi^k(\dot{\dot{k}}) .$$

## 数学代写|概率论代写Probability theory代考|Extension from Dyadic Rational Parameters to Real Parameters

Our approach to extend a given family $F$ of f.j.d.’s that is continuous in probability on the parameter set $[0,1]$ is as follows. First note that $F$ carries no more useful information than its restriction $F \mid Q_{\infty}$, where $Q_{\infty}$ is the dense subset of dyadic rationals in $[0,1]$, because the family can be recovered from the $F \mid Q_{\infty}$, thanks to continuity in probability. Hence we can first extend the family $F \mid Q_{\infty}$ to a process $Z: Q_{\infty} \times \Omega \rightarrow S$ by applying the Daniell-Kolmogorov Theorem or the Daniell-Kolmogorov-Skorokhod Theorem. Then any condition of the family $F$ is equivalent to a condition to $Z$.

In particular, in the current context, any condition on f.j.d.’s to make $F$ extendable to an a.u. continuous process $X:[0,1] \times \Omega \rightarrow S$ can be stated in terms of a process $Z: Q_{\infty} \times \Omega \rightarrow S$, with the latter to be extended by limit to the process $X$. It is intuitively obvious that any a.u. continuous process $Z: Q_{\infty} \times \Omega \rightarrow S$ is extendable to an a.u. continuous process $X:[0,1] \times \Omega \rightarrow S$, because $Q_{\infty}$ is dense $[0,1]$. In this section, we will make this precise, and we will define metrics such that the extension operation is itself a metrically continuous construction.
Definition 9.1.1. Metric space of a.u. continuous processes. Let $C[0,1]$ be the space of continuous functions $x:[0,1] \rightarrow(S, d)$, endowed with the uniform metric defined by
$$d_{C[0,1]}(x, y) \equiv \sup {t \in[0,1]} d(x(t), y(t))$$ for each $x, y \in C[0,1]$. Write $\widehat{d}{C[0,1]} \equiv 1 \wedge d_{C[0,1]}$.
Let $(\Omega, L, E)$ be an arbitrary probability space. Let $\widehat{C}[0,1]$ denote the set of stochastic processes $X:[0,1] \times(\Omega, L, E) \rightarrow(S, d)$ that are a.u. continuous on $[0,1]$. Define a metric $\rho_{\widehat{C}[0,1]}$ on $\widehat{C}[0,1]$ by
$$\rho_{\widehat{C}[0,1]}(X, Y) \equiv E \sup {t \in[0,1]} \widehat{d}\left(X_t, Y_t\right) \equiv E \widehat{d}{C[0,1]}(X, Y)$$
for each $X, Y \in \widehat{C}[0,1]$. Lemma 9.1.2 (next) says that $\left(\widehat{C}[0,1], \rho_{\widehat{C}[0,1]}\right)$ is a welldefined metric space.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Strong Law of Large Numbers

$$E\left|S_m\right| \rightarrow 0$$

$$b_m \equiv \sup k \geq q(m) E\left|S_k\right| \leq 2^{-m}$$

$$\delta(\varepsilon) \equiv \frac{\varepsilon}{2} / \eta\left(\frac{\varepsilon}{2}\right)$$

1. 根据假设，独立的 $\mathrm{rv} Z_1, Z_2, \ldots$ 有一个共同的分布 $J$ 在 $R$. 因此它们具有共同的特征 函数 $\psi$. 因此，对于每个 $k \geq 1$ ， rrv 的特征函数 $S_k$ 是 (谁) 给的
$$\psi_k \equiv \psi^k(\dot{k})$$

## 数学代写|概率论代写Probability theory代考|Extension from Dyadic Rational Parameters to Real Parameters

$$d_{C[0,1]}(x, y) \equiv \sup t \in[0,1] d(x(t), y(t))$$

$X:[0,1] \times(\Omega, L, E) \rightarrow(S, d)$ 是连续的 $[0,1]$. 定义指标 $\varphi_{\widehat{C}[0,1]}$ 在 $\widehat{C}[0,1]$ 经过
$$\rho_{\widehat{C}[0,1]}(X, Y) \equiv E \sup t \in[0,1] \hat{d}\left(X_t, Y_t\right) \equiv E \hat{d} C[0,1](X, Y)$$

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