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The canonical distance is the L2 norm of an increment of a Gaussian process. It is a pseudometric, since distinct parameters may represent equal random variables almost surely. Its metric covering number appears in the Dudley entropy integral. For standard fractional Brownian motion, $d(s,t)=|s-t{|}^H$; a normalization with twice the covariance function multiplies this distance by $\sqrt{2}$.

An $\epsilon$-net of a metric space $(T,d)$ is a subset $U\subseteq T$ such that every $x\in T$ has $d(x,u)\le \epsilon$ for some $u\in U$. Its least possible finite size is the metric covering number with closed balls. A maximal set with pairwise distances greater than $\epsilon$ is an $\epsilon$-net.

Use ${\mathbb{T}}^n=(\mathbb{R}/2\pi \mathbb{Z}{)}^n$ and the uniform metric $d(f,g)=\Vert f-g{\Vert }_{\infty }$. Let $N(\epsilon ,E)$ be the least number of radius-$\epsilon$ balls in this metric covering $E$. The Kolmogorov entropy, equivalently metric entropy, isIt is infinite if no finite cover exists; set the entropy of the empty set to zero. Changing the logarithm base or requiring centres in $E$ changes only the constants and resolution factors in the estimates below.

For an explicit appropriate norm, takeAn equivalent standard norm gives the same exponents, with different constants. We prove the two bounds for the entropy of a smooth periodic function ball separately.

For the lower bound, choose a nonnegative bump function $\phi \in C_c^{\infty }((-1,1{)}^n)$ with $0\le \phi \le 1$ and $\phi (0)=1$. Let $K$ bound all its derivatives of orders at most $p$, and choose $c>0$ with $cK\le 1$. For sufficiently small $h\le 1$, place $L\ge c_n{h}^{-n}$ translates of its scaled support disjointly on the torus, for instance by taking grid centres separated by $3h$ in a fundamental cube. Define the real functionsusing periodic extension. Disjointness means that at any point only one summand contributes. For every $|\alpha |\le p$, its derivative is bounded by $c{h}^{p-|\alpha |}K\le 1$. Thus every $f_{\sigma }$ lies in $B_{n,p}$. If $\sigma \ne \tau$, evaluate at a centre where the binary coefficients differ to obtainChoose $h=(4\epsilon /c{)}^{1/p}$ for sufficiently small $\epsilon$. The $2^L$ functions are more than $2\epsilon$ apart, so no radius-$\epsilon$ ball covers two of them. The metric covering number consequently satisfiesTo include the remaining $\epsilon <1/2$, observe that the two constant functions $1,-1$ belong to the ball and require distinct covering balls. Reducing the positive lower-bound constant extends the estimate over that fixed remaining interval.

For the upper bound, use this precise version of multivariable Jackson approximation: for every $f\in C^p({\mathbb{T}}^n)$ and integer $N\ge 1$, there exists a real trigonometric polynomial $P_N$ with $|k_j|\le N$ in every coordinate andChoose $N=\lceil (4J_{n,p}/\epsilon {)}^{1/p}\rceil$. Then the approximation error is at most $\epsilon /4$, and $\Vert P_N{\Vert }_{\infty }\le 2$. Write $P_N(x)={\sum }_{k\in \{-N,\dots ,N{\}}^n}{c}_k{e}^{ik\cdot x}$. There are $D=(2N+1{)}^n$ coefficients, and normalized integration gives $|c_k|\le \Vert P_N{\Vert }_{\infty }\le 2$.

Round their real and imaginary parts with mesh $\delta =\epsilon /(8D)$, respecting $c_{-k}=\overline{c_k}$ so the rounded polynomial remains real. Each coefficient changes by at most $2\delta$, and hence the uniform change in the polynomial is at most $2D\delta =\epsilon /4$. At most $(C/\delta {)}^{2D}$ choices are needed; allowing redundant choices only enlarges this bound. These rounded polynomials form an $\epsilon$-net for the ball, andHere $D\le C_{n,p}{\epsilon }^{-n/p}$, and the last logarithm absorbs the constant and $\log D$ for $\epsilon <1/2$. Together the bounds give

For the unit ball of $C({\mathbb{T}}^n)$ alone, take infinitely many disjoint small neighbourhoods and a height-one continuous bump supported in each. Distinct such functions have uniform distance one. Thus no finite $\epsilon$-net exists when $\epsilon <1/2$, andThe derivative bound was essential for total boundedness; mere continuity gives no common small-scale oscillation bound.

For the final composition estimate, interpret outer periodic functions on ${\mathbb{T}}^m$ by their $2\pi$-periodic extensions to ${\mathbb{R}}^m$. Since $p\ge 1$, each $f\in B_{m,p}$ satisfiesby integrating its first partial derivatives along coordinate segments. This uniform Lipschitz continuity controls errors in the inner functions.

Choose an outer net of radius $\eta =\epsilon /2$ for $B_{m,p}$ and an inner net of radius $\delta =\epsilon /(2m)$ for $E$. For each $g=f(u_1,\dots ,u_m)$ choose corresponding approximants $f_0,v_1,\dots ,v_m$. No derivative bound for $f_0$ is needed: use the derivative bound of $f$ before replacing the outer function. Pointwise,All composed centres are continuous. Multiplying the numbers of outer and inner choices gives the entropy of Lipschitz compositions inequalityApply the proved upper bound with dimension $m$, and the assumed estimate for $E$. Fixed rescalings of $\epsilon$ preserve its exponent and change $\log (1/\epsilon )$ by only an additive constant. Therefore

The equal-variance form of Slepian's lemma is the following. If $U,V$ are centered vectors with multivariate normal distributions, $\mathbb{E}{U}_i^2=\mathbb{E}{V}_i^2$ for every $i$, and $\mathbb{E}{U}_i{U}_j\ge \mathbb{E}{V}_i{V}_j$ whenever $i\ne j$, thenfor every threshold vector $(u_i)$. In particular $\mathbb{P}({\max }_i{U}_i\le u)\ge \mathbb{P}({\max }_i{V}_i\le u)$. No nonsingularity of the covariance matrices is required.

Here the pointwise variances $2t^{2H}$ and $2t^{2H^{\prime }}$ differ. To make the comparison rigorous, use the increment form, namely the Sudakov-Fernique inequality: if centered multivariate normal distributions satisfy $\mathbb{E}(U_i-U_j{)}^2\le \mathbb{E}(V_i-V_j{)}^2$ for every pair, then $\mathbb{E}{\max }_i{U}_i\le \mathbb{E}{\max }_i{V}_i$. Unlike the preceding form of Slepian's lemma, this statement has no equal-pointwise-variance hypothesis.

For completeness, the increment form follows by Gaussian integration by parts. Make $U,V$ independent, define $Z_r=\sqrt{1-r}U+\sqrt{r}V$ for $0<r<1$, and letThe $p_i$ are softmax function weights, with $p_i\ge 0$ and ${\sum }_i{p}_i=1$. Direct differentiation gives ${\partial }_i{F}_{\tau }=p_i$ and ${\partial }_{ij}{F}_{\tau }=\tau (p_i{1}_{i=j}-p_i{p}_j)$. The Gaussian integration by parts identity therefore yieldsThe last summands are nonnegative because $D_{ii}+D_{jj}-2D_{ij}=\mathbb{E}(V_i-V_j{)}^2-\mathbb{E}(U_i-U_j{)}^2$. Sinceintegration in $r$ and the limit $\tau \to \infty$ prove the Sudakov-Fernique inequality. All derivatives needed here are bounded for a fixed $\tau$. Singular covariance matrices can be handled by adding small independent centered normal noise with the same variance to both vectors, proving the identity for the nonsingular vectors, and letting the noise variance decrease to zero.

Use the printed normalization of fractional Brownian motion, which omits the conventional factor $1/2$. Its increment variance isFor $s,t\in [0,1]$ and $H>H^{\prime }$, $|t-s{|}^{2H}\le |t-s{|}^{2H^{\prime }}$. Enumerate $T$ by increasing finite subsets $T_n$ whose union is $T$, with $0\in T_n$. The Sudakov-Fernique inequality givesBoth Gaussian processes vanish at zero almost surely. Thus these maxima are nonnegative and increase with $n$; monotone convergence theorem givesTheir continuous modifications agree with the original Gaussian processes simultaneously on the countable set $T$, after discarding a single null event. Continuity and density of the rational parameters make each rational supremum equal to the supremum over $[0,1]$ for these modifications.

The expected suprema are finite. Indeed, the canonical distance of $Y_{H^{\prime }}$ is $d(s,t)=\sqrt{2}|t-s{|}^{H^{\prime }}$. A Euclidean grid supplies a closed-ball metric covering number bound $N(\epsilon )\le 1+\lceil (\sqrt{2}/\epsilon {)}^{1/H^{\prime }}\rceil$ for $0<\epsilon \le \sqrt{2}$. The Dudley entropy integral bound for an anchored centered separable Gaussian process isThe integral is finite since its integrand grows at most like a constant times $\sqrt{1+\log (1/\epsilon )}$ near zero. The same argument applies to $X_H$.

For any real path vanishing at zero,A centered Gaussian process has the same probability law as its negative. Combining this symmetry with the increment comparison gives the stronger boundIn particular the requested comparison isThe common normalization factor $\sqrt{2}$ affects both sides equally. The comparison is valid for every $0<H^{\prime }<H<1$, including the equality cases $s=t$ or $|s-t|=1$ in the increment bounds.