## 统计代写|随机过程代写stochastic process代考|STAT7004

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## 统计代写|随机过程代写stochastic process代考|Rolling Up Our Sleeves: Chaining in the Simplex

The bound (2.34) seems to be genuinely better than the bound ( $2.38$ ) because when going from (2.34) to (2.38) we have used the somewhat brutal inequality:
$$\sup {t \in T} \sum{n \geq 0} 2^{n / 2} d\left(t, T_n\right) \leq \sum_{n \geq 0} 2^{n / 2} \sup {t \in T} d\left(t, T_n\right) .$$ The method leading to the bound (2.34) is probably the most important idea of this work. The fact that it appears now so naturally does not reflect the history of the subject, but rather that the proper approach is being used. When using this bound, we will choose the sets $T_n$ in order to minimize the right-hand side of (2.34) instead of choosing them as in (2.36). As we will demonstrate later, this provides essentially the best possible bound for $\mathrm{Esup}{t \in T} X_t$. It is remarkable that despite the fact that this result holds in complete generality, it is a non-trivial task to find sets $T_n$ witnessing this, even in very simple situations. In the present situation, we perform this task by an explicit construction for the set $T$ of (2.49).
Proposition 2.6.1 There exist sets $T_n \subset \mathbb{R}^m$ with card $T_n \leq N_n$ such that
$$\sup {t \in T} \sum{n \geq 0} 2^{n / 2} d\left(t, T_n\right) \leq L \sqrt{\log m}\left(=L E \sup _{t \in T} X_t\right) .$$
Of course here $d$ is the Euclidean distance in $\mathbb{R}^m$. The reader may try to find these sets herself before reading the rest of this section, as there seems to be no better way to get convinced of the depth of the present theory. The sets $T_n$ are not subsets of $T$. Please figure out by yourself how to correct this. ${ }^{12}$

Lemma 2.6.2 For each $t \in T$, we can find a sequence $(p(n, t)){n \geq 0}$ of integers $0 \leq p(n, t) \leq 2 n$ with the following properties: $$\begin{gathered} \sum{n \geq 0} 2^{n-p(n, t)} \leq L, \ \forall n \geq 0, p(n+1, t) \leq p(n, t)+2, \ \text { card }\left{i \leq m ; t_i \geq 2^{-p(n, t)}\right}<2^n . \end{gathered}$$

## 统计代写|随机过程代写stochastic process代考|Admissible Sequences of Partitions

The idea behind the bound (2.34) admits a technically more convenient formulation. ${ }^{14}$

Definition 2.7.1 Given a set $T$, an admissible sequence is an increasing sequence $\left(\mathcal{A}n\right){n \geq 0}$ of partitions of $T$ such that card $\mathcal{A}_n \leq N_n$, i.e., card $\mathcal{A}_0=1$ and card $\mathcal{A}_n \leq$ $2^{2^n}$ for $n \geq 1$

By an increasing sequence of partitions, we mean that every set of $\mathcal{A}{n+1}$ is contained in a set of $\mathcal{A}_n$. Admissible sequences of partitions will be constructed recursively, by breaking each element $C$ of $\mathcal{A}_n$ into at most $N_n$ pieces, obtaining then a partition $\mathcal{A}{n+1}$ of $T$ consisting of at most $N_n^2 \leq N_{n+1}$ pieces.

Throughout the book, we denote by $A_n(t)$ the unique element of $\mathcal{A}_n$ which contains $t$. The double exponential in the definition of $N_n$ (see (2.29)) occurs simply since for our purposes the proper measure of the “size” of a partition $\mathcal{A}$ is $\log \operatorname{card} \mathcal{A}$. This double exponential ensures that “the size of the partition $\mathcal{A}_n$ doubles at every step”. This offers a number of technical advantages which will become clear gradually.

Theorem 2.7.2 (The Generic Chaining Bound) Under the increment condition (2.4) (and if $\mathrm{E} X_t=0$ for each $t$ ), then for each admissible sequence $\left(\mathcal{A}n\right)$ we have $$\mathrm{E} \sup {t \in T} X_t \leq L \sup {t \in T} \sum{n \geq 0} 2^{n / 2} \Delta\left(A_n(t)\right) .$$
Here as always, $\Delta\left(A_n(t)\right)$ denotes the diameter of $A_n(t)$ for $d$. One could think that (2.54) could be much worse than (2.34), but it will turn out that this is not the case when the sequence $\left(\mathcal{A}_n\right)$ is appropriately chosen.

Proof We may assume $T$ to be finite. We construct a subset $T_n$ of $T$ by taking exactly one point in each set $A$ of $\mathcal{A}_n$. Then for $t \in T$ and $n \geq 0$, we have $d\left(t, T_n\right) \leq$ $\Delta\left(A_n(t)\right)$ and the result follows from (2.34).

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Rolling Up Our Sleeves: Chaining in the Simplex

$$\sup t \in T \sum n \geq 02^{n / 2} d\left(t, T_n\right) \leq \sum_{n \geq 0} 2^{n / 2} \sup t \in T d\left(t, T_n\right)$$

$$\sup t \in T \sum n \geq 02^{n / 2} d\left(t, T_n\right) \leq L \sqrt{\log m}\left(=L E \sup _{t \in T} X_t\right)$$

## 统计代写|随机过程代写stochastic process代考|Admissible Sequences of Partitions

$\operatorname{Esup} t \in T X_t \leq L \sup t \in T \sum n \geq 02^{n / 2} \Delta\left(A_n(t)\right)$

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