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Let $(X_t{)}_{t\in T}$ be a centered Gaussian process with a separable version: its relevant suprema are captured by a countable dense index subset. Give $T$ the canonical pseudometric of a Gaussian process $d(s,t)=\Vert X_s-X_t{\Vert }_2$, and assume it is totally bounded. With metric covering number $N(\epsilon ,T,d)$ and diameter $D$, the Dudley entropy integral boundsfor a universal constant $C$. Finiteness of the integral gives a finite expected supremum. For an absolute supremum, include an anchor: ${\sup }_t|X_t|\le |X_{t_0}|+{\sup }_t|X_t-X_{t_0}|$. The anchor term cannot generally be omitted.

The proof builds successively finer nets. Write each increment as a telescoping sum between neighboring net scales, bound each finite Gaussian maximum by its scale times the square root of the logarithmic net size, and sum over scales. This becomes the displayed entropy integral.^[ref]

The version assumption is substantive. If $U$ is uniform on $[0,1]$ and $X_t=1_{\{U=t\}}$, every finite vector is zero almost surely and is therefore a degenerate Gaussian vector; the canonical pseudometric is identically zero. Yet ${\sup }_t{X}_t=1$ for every outcome. Its all-zero separable modification has supremum zero. Finite-dimensional laws alone do not control the supremum of an arbitrary uncountable version.

For a Gaussian random vector $X\sim N(m,\Sigma )$ and an $L$-Lipschitz function $f$ in the Euclidean norm, put $v=L^2\Vert \Sigma {\Vert }_{\mathrm{op}}$. If $v>0$, the Gaussian concentration inequality givesand thereforeWrite $X=m+{\Sigma }^{1/2}G$ with a standard Gaussian random vector $G$. The composed function has Lipschitz constant at most $L\Vert \Sigma {\Vert }_{\mathrm{op}}^{1/2}$. The Gaussian logarithmic Sobolev inequality and Herbst argument give the exponential bound, and the Chernoff bound supplies the tails.^[ref]

For example, $f(x)=\Vert x\Vert$ has $L=1$, so its deviations depend on the largest covariance eigenvalue rather than directly on the number of coordinates. The center $\mathbb{E}\Vert X\Vert$ can still grow with dimension. If $v=0$, $f(X)$ is constant almost surely and every positive deviation has probability zero.

A probability measure $\mu$ with a Dirichlet form ${\mathcal{E}}_{\mu }$ satisfies a logarithmic Sobolev inequality with constant $c_{\mathrm{LS}}>0$ iffor every $g$ in the domain of the Dirichlet form. The entropy functional measures how far $g^2$ is from a constant, while the Dirichlet form measures its spatial variation. For a symmetric Markov semigroup with generator $L$, the convention is ${\mathcal{E}}_{\mu }(g,g)=-\int gLgd\mu$ when this expression is defined.

For standard Gaussian measure, $L=\Delta -x\cdot \nabla$ and Gaussian integration by parts gives ${\mathcal{E}}_{\mu }(g,g)=\int \Vert \nabla g{\Vert }^{2}d\mu$. The Gaussian logarithmic Sobolev inequality has optimal constant two in this convention. Rescaling the Dirichlet form rescales the constant inversely, so a numerical constant needs its energy convention. The logarithmic Sobolev inequality implies hypercontractivity and also supports concentration inequalities through the Herbst argument.^[ref]

// Target: analysis.bigb

Ramon van Handel authored notes on probability theory in high dimensions connecting probability theory, analysis and geometry. The notes develop concentration of measure, Gaussian processes and related inequalities through common mechanisms such as tensorization of entropy. This viewpoint explains why a function of many independent inputs can be tightly concentrated even when the full input space has very high dimension.^[ref]

A useful precise version of Talagrand's one-sided bounded differences inequality assumes that independent random variables $X_i$ enter a measurable function $f$ for which there are nonnegative coordinate weights $c_i(x)$ satisfying the global comparisonPut $Z=f(X)$ and $V=\mathbb{E}{\sum }_i{c}_i(X{)}^2$. When $0<V<\infty$ and $Z$ is integrable,The global comparison is stronger than bounding separate coordinate changes at one input. This distinction matters for a lower tail.^[ref]

For a differentiable convex function on $[0,1{]}^n$, the supporting-gradient inequality verifies the comparison with $c_i(x)=|{\partial }_{i}f(x)|$. Thus $V=\mathbb{E}\Vert \nabla f(X){\Vert }^2$ controls the lower tail. Applying the theorem to the negative of a concave function gives its upper tail. For example, $f(x)={\sum }_i{x}_i$ has $V=n$, giving $e^{-t^2/(2n)}$ as a valid, though conservative, lower-tail bound.

A precise sum-cost form of tensorization of a transport-entropy inequality is as follows. Let ${\mu }_i$ be probability measures on Polish spaces and $c_i$ nonnegative continuous transport cost functions. Suppose for every probability measure ${\nu }_i$ thatThen for the product measure $\mu ={⨂}_i{\mu }_i$ and sum cost $c(x,y)={\sum }_i{c}_i(x_i,y_i)$,To understand the mechanism first for finite spaces, reveal the coordinates under $\nu$ in order. Given the previous coordinates $x_{<i}$, couple the conditional law ${\nu }_i(\cdot \mid x_{<i})$ to ${\mu }_i$. Choosing these couplings successively preserves the product measure as the second marginal. The total expected cost is at most $C{\sum }_i{\mathbb{E}}_{\nu }D({\nu }_i(\cdot \mid X_{<i})\Vert {\mu }_i)$, which equals $CD(\nu \Vert \mu )$ by the chain rule for relative entropy. Approximate optimal couplings give the infimum. This statement spells out the cost and constant; it does not identify every different distance-based normalization with the same formula.^[ref]

A transport-entropy inequality compares the cost of moving one probability measure to another with their relative entropy. For a nonnegative transport cost function $c$ and reference probability measure $\mu$, definewhere $\Pi (\nu ,\mu )$ consists of couplings with those marginals. One common convention is ${\mathcal{T}}_c(\nu ,\mu )\le CD(\nu \Vert \mu )$ for every probability measure $\nu$; the orientation of the relative entropy matters.

For a metric $d$, with the measures having finite first moments, the $T_1(C)$ convention is Wasserstein distance $W_1(\nu ,\mu {)}^2\le 2CD(\nu \Vert \mu )$, while $T_2(C)$ replaces $W_1$ by the second Wasserstein distance $W_2$. The first convention bounds fluctuations of all 1-Lipschitz functions: $\log {\mathbb{E}}_{\mu }{e}^{\lambda (f-{\mathbb{E}}_{\mu }f)}\le C{\lambda }^2/2$. The choice of a squared distance or an unsquared transport cost function is part of the statement, rather than a universal normalization.^[ref]