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For a real-valued stochastic process $(X_t)$ with finite second moments, write $m(t)=\mathbb{E}{X}_t$. Its covariance function isThe diagonal gives the variance at each time; off-diagonal values record linear dependence between times. For the independent standard normal random variables above, $C_X(m,n)=1_{\{m=n\}}$, whereas the shared Gaussian process $Y_n=Z$ has $C_Y(m,n)=1$ for all $m,n$. For a fair simple random walk on the integer line, $C(m,n)=\min (m,n)$; Brownian motion has the continuous-time counterpart $C(s,t)=\min (s,t)$. These formulas quantify how much random history the observations share.

Every covariance function is a positive-semidefinite kernel. Indeed, for any real coefficients $a_1,\dots ,a_k$,Thus a proposed covariance function must make every finite covariance matrix positive semidefinite. For a Gaussian process, the mean function and the covariance function determine every finite-dimensional distribution, including degenerate ones such as the shared process. Conversely, any real positive-semidefinite kernel satisfying $C(s,t)=C(t,s)$, together with a specified mean function, defines consistent multivariate normal distributions; the Kolmogorov extension theorem then constructs a Gaussian process. For a general stochastic process, covariance alone does not determine the full law. The exceptional completeness of this description is what makes Gaussian process regression possible with a mean and a covariance model.

Standard one-dimensional Brownian motion is a real process with $B_0=0$, continuous paths almost surely, and increments $B_t-B_s\sim N(0,t-s)$ independent of the past at time $s$, for $0\le s<t$. These conditions determine its entire finite-dimensional distribution family.

The Wiener theorem asserts that such a process exists, equivalently that there is a unique Wiener measure on $C_0([0,\infty ),\mathbb{R})$ under which the coordinate process has these properties. To construct it, prescribe centered multivariate normal laws with covariance $\mathbb{E}{B}_s{B}_t=\min (s,t)$. This covariance is positive semidefinite becauseThe laws are consistent under removing coordinates, so the Kolmogorov extension theorem constructs the process. Normal increments have fourth moment $\mathbb{E}|B_t-B_s{|}^4=3|t-s{|}^2$. The Kolmogorov continuity theorem therefore supplies a continuous modification, retaining all the prescribed finite-dimensional laws and hence the independent normal increments. Apply this on every compact time interval, taking consistent versions; continuity and equality at rational times make them agree on overlaps. The resulting law on continuous-path space is unique because evaluations at rational times generate its Borel sigma-algebra. This is a proof sketch of the existence theorem, independent of the random-walk limit in Question 5.

First construct a centred Gaussian process on the countable dyadic set with covariance function $\mathbb{E}(B_s{B}_t)=\min (s,t)$. These consistent finite-dimensional normal distributions exist because, for coefficients $a_j$,the Kolmogorov extension theorem realizes them on one probability space. Every increment has the normal distribution $N(0,|t-s|)$, so for every finite $p\ge 1$,Use the preceding estimate with $\beta =1/2$, $p>2$ and $0<\alpha <1/2-1/p$. It yields an almost surely Hölder continuous extension from the dense dyadic set to $[0,1]$. Approximate arbitrary times by dyadic times; continuity of the paths and of the Gaussian covariance function preserves all finite-dimensional laws. Disjoint increments are jointly Gaussian with zero cross-covariances, hence independent, with the required variances. Together with $B_0=0$ and continuity, this constructs Brownian motion.

For any $\alpha <1/2$, choose $p$ large enough. Taking a countable increasing sequence of exponents tending to $1/2$ gives simultaneous Hölder regularity of every order $\alpha <1/2$ on a common probability-one set; the same argument on each compact time interval gives local Brownian Hölder regularity. This method does not give the endpoint exponent. In fact that endpoint fails: if a path had a finite $1/2$-Hölder seminorm $K$, all its $2^n$ adjacent normalized increments at level $n$ would have magnitude at most $K$. These are independent standard normal random variables, so for a fixed integer $m$ the probability that their maximum is at most $m$ is $\mathbb{P}(|Z|\le m{)}^{2^n}\to 0$. The event of a finite $K$ is contained in the union over $m$ of events with this bound at every level, each of probability zero. Thus Brownian paths are not $1/2$-Hölder continuous on $[0,1]$ almost surely.

For a probability distribution $m$ on $\mathbb{R}$, take the canonical sample space $\Omega ={\mathbb{R}}^{\mathbb{N}}$, the product of the Borel $\sigma$-algebras, and the product measure $\mathbb{P}=m^{\otimes \mathbb{N}}$. The coordinate functions $X_j(\omega )={\omega }_j$ have joint cylinder probabilitiesThe Kolmogorov extension theorem supplies this measure from the consistent finite-dimensional distributions, and uniqueness follows from the generating cylinder sets. Thus the coordinates are independent and identically distributed; the same construction works for a given standard Borel state space.

The one-sided Bernoulli shift is $T({\omega }_1,{\omega }_2,\dots )=({\omega }_2,{\omega }_3,\dots )$. It is measurable and preserves all cylinder probabilities. The uniqueness theorem for probability measures extends this to the product $\sigma$-algebra, proving measure preservation.

To prove ergodicity, let $A$ be invariant modulo null sets and approximate it in probability by an event $C$ depending on only the first $r$ coordinates, with $\mathbb{P}(A△C)<ϵ$. Such approximations follow from the fact that finite-coordinate cylinder events generate the product $\sigma$-algebra. For $n\ge r$, independence gives $\mathbb{P}(C\cap T^{-n}C)=\mathbb{P}(C{)}^2$. Invariance and measure preservation giveAlso $|\mathbb{P}(A{)}^2-\mathbb{P}(C{)}^2|\le 2ϵ$. Letting $ϵ\downarrow 0$ proves $\mathbb{P}(A)=\mathbb{P}(A{)}^2$, so $\boxed{\mathbb{P}(A)\in \{0,1\}}$. This is exactly ergodicity.

There exists a probability measure on $C_0([0,\infty ),\mathbb{R})$ under which the coordinate process is standard Brownian motion, and its distribution there is unique. Construct consistent centered Gaussian finite-dimensional laws with covariance $\min (s,t)$ and use the Kolmogorov extension theorem. The fourth increment moment $3|t-s{|}^2$ supplies a continuous modification by the Kolmogorov continuity theorem. Its independent increments and normal increment distributions are unchanged. Uniqueness follows because coordinate evaluations at rational times generate the Borel sigma-algebra of continuous-path space.