# 统计代写|随机过程代写stochastic process代考|STATS217

#### Doug I. Jones

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## 统计代写|随机过程代写stochastic process代考|Partitioning Schemes

In this section, we use functionals satisfying the growth condition to construct admissible sequences of partitions. The basic result is as follows:

Theorem 2.9.1 Assume that there exists on $T$ a functional $F$ which satisfies the growth condition of Definition $2.8 .3$ with parameters $r$ and $c^$. Then ${ }^{20}$ $$\gamma_2(T, d) \leq \frac{L r}{c^} F(T)+\operatorname{Lr} \Delta(T) .$$
This theorem and its generalizations form the backbone of this book. The essence of this theorem is that it produces (by actually constructing them) a sequence of partitions that witnesses the inequality (2.81). For this reason, it could be called “the fundamental partitioning theorem”.

Exercise 2.9.2 Consider a metric space $T$ consisting of exactly two points. Prove that the functional given by $F(H)=0$ for each $H \subset T$ satisfies the growth condition of Definition $2.8 .3$ for $r=8$ and any $c^>0$. Explain why we cannot replace (2.81) by the inequality $\gamma_2(T, d) \leq L r F(T) / c^$.

Let us first stress the following trivial fact (connected to Exercise $2.5 .9$ (a)). It will be used many times. The last statement of (a) is particularly useful.

(a) We pick the points $t_{\ell}$ recursively with $d\left(t_{\ell}, t_{\ell^{\prime}}\right) \geq a$ for $\ell^{\prime}<\ell$. By hypothesis, the balls of radius $a$ centered on the previously constructed points do not cover the space if there are $<N$ of them so that the construction continues until we have constructed $N$ points.
(b) You can either view this as a reformulation of (a) or argue directly that when $m$ is taken as large as possible the balls $B\left(t_{\ell}, a\right)$ cover $T$.
(c) If $T$ is covered by sets $\left(B_{\ell^{\prime}}\right){\ell^{\prime}} \leq N_n$, by the pigeon hole principle, at least two of the points $t{\ell}$ must fall into one of these sets, which therefore cannot be a ball of radius $<a / 2$.

The admissible sequence of partitions witnessing (2.81) will be constructed by recursive application of the following basic principle.

## 统计代写|随机过程代写stochastic process代考|The Majorizing Measure Theorem

Consider a Gaussian process $\left(X_t\right){t \in T}$, that is, a jointly Gaussian family of centered r.v.s indexed by $T$. We provide $T$ with the canonical distance $$d(s, t)=\left(\mathrm{E}\left(X_s-X_I\right)^2\right)^{1 / 2}$$ Recall the functional $\gamma_2$ of Definition 2.7.3. Theorem 2.10.1 (The Majorizing Measure Theorem) For a universal constant $L$, it holds that $$\frac{1}{L} \gamma_2(T, d) \leq \mathrm{E} \sup {t \in T} X_t \leq L \gamma_2(T, d)$$
The reason for the name is explained in Sect. 3.1. We will meditate on this statement in Sect. 2.12. We will spend much time trying to generalize this theorem to other classes of processes. To link the statements of these generalizations with that of (2.114), it may be good to reformulate the lower bound $\gamma_2(T, d) \leq L E \sup {t \in T} X_t$ in the following general terms: The control from above of $\mathrm{E} \sup {t \in T} X_t$ implies the existence of a
“small” sequence of admissible partitions of $T$.
The right-hand side inequality in (2.114) is Theorem 2.7.11. To prove the lower bound, we will use Theorem $2.9 .1$ and the functional
$$F(H)=\mathrm{E} \sup {t \in H} X_t:=\sup {H^* \subset H, H^* \text { finite }} \mathrm{E} \sup _{t \in H^} X_t$$ For this, we need to prove that this functional satisfies the growth condition with $c^$ a universal constant and to bound $\Delta(T)$. We strive to give a proof that relies on general principles and lends itself to generalizations.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Partitioning Schemes

Igamma_2(T, d) \eq \frac ${L$ r $}\left{c^{\wedge}\right} F(T)+l o p e r a t o r n a m e{L$

(a) 我们挑选点 $t_{\ell}$ 递归地 $d\left(t_{\ell}, t_{\ell^{\prime}}\right) \geq a$ 为了 $\ell^{\prime}<\ell$. 根 据假设，半径为 $a$ 如果有，则以先前构造的点为中心不 覆盖空间 $<N$ 他们的建设，以便建设继续，直到我们已 经建设 $N$ 点。
(b) 你可以将其视为对 (a) 的重新表述，也可以直接争辩 说当 $m$ 尽可能大的球 $B\left(t_{\ell}, a\right)$ 覆盖 $T$.
(c) 如果 $T$ 被集合覆盖 $\left(B_{\ell^{\prime}}\right) \ell^{\prime} \leq N_n$ ，根据䴓巣原理， 至少有两个点 $t \ell$ 必须属于这些集合之一，因此不能是半 径为 $<a / 2$.

## 统计代写|随机过程代写stochastic process代考|The Majorizing Measure Theorem

$$d(s, t)=\left(\mathrm{E}\left(X_s-X_I\right)^2\right)^{1 / 2}$$

$$\frac{1}{L} \gamma_2(T, d) \leq \operatorname{E} \sup t \in T X_t \leq L \gamma_2(T, d)$$

$\operatorname{Esup} t \in T X_t$ 意味着存在一个
“小”序列的可接受的分区 $T$.
(2.114) 右边的不等式是定理 2.7.11。为了证明下界，我 们将使用定理 $2.9 .1$ 和功能
$F(H)=\backslash m a t h r m{E} \backslash \sup \left{t\right.$ in $\left.H^{\prime}\right} X_{-} t:=\mid \sup \left{H^{\wedge} * \mid\right.$ subset

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