Proof of Existence of Independent Sequences with Prescribed Distributions

Throughout, $\mathbb{N}$ denotes the natural numbers, $\mathbb{R}$ the real numbers, and $\lambda$ Lebesgue measure. For $t\ge0$, $\lfloor t\rfloor$ denotes the greatest natural number $\le t$, which exists by the Archimedean property. We use repeatedly that the closed rays $(-\infty,t]$, $t\in\mathbb{R}$, together with $\mathbb{R}$, form a $\pi$-system generating the Borel $\sigma$-algebra: every open interval with rational endpoints is obtained from rays by countable set operations, and every open set is a countable union of such intervals by density of the rationals.

Step 0 (the space). Take $\Omega=(0,1)$, $\mathcal{F}=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq(0,1)\}$, and $P$ the restriction of $\lambda$. $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$ (complements within $(0,1)$ and countable unions of Borel subsets of $(0,1)$ are again Borel subsets of $(0,1)$), $P$ inherits countable additivity from $\lambda$ (Measure, Measure Space, and Probability Measure), and $P(\Omega)=\lambda((0,1))=1$ since $\lambda$ assigns each interval its length. So $(\Omega,\mathcal{F},P)$ is a probability space.

Step 1 (binary digits). For $j\ge1$ define $d_j:\Omega\to\mathbb{R}$ by $d_j(\omega)=\lfloor 2^{j}\omega\rfloor-2\lfloor 2^{j-1}\omega\rfloor$. Writing $t=2^{j-1}\omega$: from $2\lfloor t\rfloor\le 2t<2\lfloor t\rfloor+2$ we get $\lfloor 2t\rfloor\in\{2\lfloor t\rfloor,\,2\lfloor t\rfloor+1\}$, so $d_j$ takes only the values $0$ and $1$. For $0\le\ell<2^{j}$ let $I_{j,\ell}=[\ell 2^{-j},(\ell+1)2^{-j})\cap(0,1)$ (the level-$j$ dyadic intervals). On $I_{j,\ell}$ we have $\lfloor 2^{j}\omega\rfloor=\ell$; hence each $d_i$ with $i\le j$ is constant on every level-$j$ interval, and $d_j=1$ on $I_{j,\ell}$ exactly when $\ell$ is odd. In particular each $d_j$ is a simple function with Borel level sets, hence a random variable. By induction on $L$, the prefix sums satisfy

(Digit patterns.) Fix distinct indices $j_1<\dots<j_s$ and values $e_1,\dots,e_s\in\{0,1\}$. We claim

The event is a union of level-$j_s$ dyadic intervals, and we count them by induction on $s$. For $s=1$: $d_{j_1}=e_1$ on exactly $2^{\,j_1-1}$ of the $2^{\,j_1}$ level-$j_1$ intervals (those of the correct parity of $\ell$). Inductive step: suppose the event for $e_1,\dots,e_{s-1}$ is a union of exactly $2^{\,j_{s-1}-(s-1)}$ level-$j_{s-1}$ intervals. Every level-$j'$ interval with $j'<j_s$ is the disjoint union of its $2^{\,j_s-j'}$ level-$j_s$ subintervals, and $d_{j_s}$ equals $e_s$ on exactly half of the level-$j_s$ subintervals of any level-$(j_s-1)$ interval (its two subintervals carry $d_{j_s}=0$ and $d_{j_s}=1$ respectively), hence on exactly half of the level-$j_s$ subintervals of any level-$j'$ interval with $j'<j_s$. The count is therefore $2^{\,j_{s-1}-(s-1)}\cdot\tfrac12\cdot 2^{\,j_s-j_{s-1}}=2^{\,j_s-s}$. Each level-$j_s$ interval has $P$-measure $2^{-j_s}$ (its length; the leftmost is the interval $(0,2^{-j_s})$, of the same length), so the event has probability $2^{\,j_s-s}\cdot 2^{-j_s}=2^{-s}$. In particular $P(d_j=0)=P(d_j=1)=1/2$.

Step 2 (independent uniforms). Every positive natural number is uniquely of the form $2^{k-1}(2i-1)$ with $i,k\ge1$ (extract the largest power of two dividing the number; uniqueness by parity), so $\sigma_k(i)=2^{k-1}(2i-1)$ defines a bijection of index pairs onto the positive naturals, with disjoint ranges for distinct $k$. Define $s^{(k)}_L=\sum_{i=1}^{L}2^{-i}d_{\sigma_k(i)}$ and $U_k=\sup_L s^{(k)}_L$, the supremum existing since the $s^{(k)}_L$ are nondecreasing in $L$ and bounded by $1$ (Least Upper Bound Property of the Real Numbers); $U_k$ is also the limit of the $s^{(k)}_L$. Each $s^{(k)}_L$ is a simple function, and $U_k$ is a random variable with values in $[0,1]$, since $\{U_k>t\}=\bigcup_L\{s^{(k)}_L>t\}\in\mathcal{F}$ and the rays $(t,\infty)$ generate $\mathcal{B}(\mathbb{R})$ (generator criterion of Measurable Function and Real-Valued Measurable Function). Since $s^{(k)}_L\uparrow U_k$ pointwise, $\{U_k\le t\}=\bigcap_L\{s^{(k)}_L\le t\}$, a decreasing intersection.

(Joint distribution functions.) Fix $p\ge1$, distinct $k_1,\dots,k_p$, and $t_1,\dots,t_p\in\mathbb{R}$. By continuity from above of the finite measure $P$ (complement an increasing union and use countable additivity),

For fixed $L$, each event $\{s^{(k_a)}_L\le t_a\}$ is the disjoint union, over the patterns $(e_1,\dots,e_L)\in\{0,1\}^L$ with $\sum_{i\le L}2^{-i}e_i\le t_a$, of the pattern events $\{d_{\sigma_{k_a}(i)}=e_i\ \text{for}\ i\le L\}$. The digit index sets $\{\sigma_{k_a}(i):i\le L\}$, $a=1,\dots,p$, are pairwise disjoint, so expanding the intersection over $a$ yields a disjoint union of combined pattern events on $pL$ distinct digits, each of probability $2^{-pL}$ by Step 1 — the product of the individual pattern probabilities $2^{-L}$. Summing over admissible pattern combinations factors as a product of sums, with the finite product notation:

Letting $L\to\infty$ on both sides,

(Uniformity.) Let $t\in[0,1)$. The map $(e_1,\dots,e_L)\mapsto\sum_{i\le L}2^{\,L-i}e_i$ is a bijection onto $\{0,1,\dots,2^L-1\}$ (uniqueness of binary representations of these integers, by induction on $L$), and $\sum_{i\le L}2^{-i}e_i=2^{-L}\sum_{i\le L}2^{\,L-i}e_i$; hence the number of patterns with $\sum_{i\le L}2^{-i}e_i\le t$ is $\lfloor 2^{L}t\rfloor+1$, and by Step 1,

Letting $L\to\infty$: $P(U_k\le t)=t$ for all $t\in[0,1)$; trivially $P(U_k\le t)=0$ for $t<0$ and $=1$ for $t\ge1$. In particular $P(U_k=0)\le P(U_k\le0)=0$, and $P(U_k=1)\le 1-P(U_k\le t)=1-t$ for every $t\in[0,1)$, so $P(U_k=1)=0$ and $P(U_k\in(0,1))=1$.

(Independence of the uniforms.) We upgrade $(*)$ to all Borel sets one coordinate at a time, by double induction on the family size $p$ and on the number $a_0\in\{0,\dots,p\}$ of coordinates already upgraded; the claim is that for all Borel $B_1,\dots,B_{a_0}$ and all reals $t_{a_0+1},\dots,t_p$,

The case $a_0=0$ is $(*)$. For the step $a_0\to a_0+1$, fix the other data and compare, as functions of $B\in\mathcal{B}(\mathbb{R})$, the two finite measures given by the left and right sides with $B$ in the $(a_0+1)$-st slot (countable additivity as usual via disjoint preimages). They agree on the closed rays by the induction hypothesis at $a_0$, and on $B=\mathbb{R}$ by the identity for the subfamily with $k_{a_0+1}$ removed (induction on $p$). The class where two finite measures of equal total mass agree contains the whole space and is closed under proper differences and increasing countable unions (additivity, finiteness, continuity from below), i.e. it is a $\lambda$-system; it contains the generating $\pi$-system of closed rays together with $\mathbb{R}$, so by Dynkin's Pi-Lambda Theorem it is all of $\mathcal{B}(\mathbb{R})$. With $a_0=p$ we conclude, for all Borel $B_a$,

taking $B_a=\mathbb{R}$ for indices outside a subset covers all the product identities required, so the family $(U_k)_{k\ge1}$ is independent.

Step 3 (quantile transforms). Fix $m\in\mathbb{N}$ and set $F_m(t)=\nu_m((-\infty,t])$. Then $F_m$ is nondecreasing (monotonicity of measures), right-continuous ($F_m(t)=\lim_n F_m(t+1/n)$ by continuity from above of the finite measure $\nu_m$ along $(-\infty,t+1/n]\downarrow(-\infty,t]$), and has limits $0$ at $-\infty$ and $1$ at $+\infty$ (continuity from above along $(-\infty,-n]\downarrow\emptyset$, continuity from below along $(-\infty,n]\uparrow\mathbb{R}$, and monotonicity). For $u\in(0,1)$ define

a real number: the set is nonempty (since $F_m(t)\to1>u$) and bounded below (since $F_m(t)\to0<u$), so the greatest lower bound exists via Least Upper Bound Property of the Real Numbers. Key equivalence: for $u\in(0,1)$ and $t\in\mathbb{R}$,

Indeed, if $u\le F_m(t)$ then $t$ lies in the set, so $q_m(u)\le t$. Conversely, the set $\{t:F_m(t)\ge u\}$ is upward closed by monotonicity and has infimum $q_m(u)$, so it contains $(q_m(u),\infty)$; right-continuity gives $F_m(q_m(u))=\lim_n F_m(q_m(u)+1/n)\ge u$, and monotonicity extends this to all $t\ge q_m(u)$.

Consequently $\{u\in(0,1):q_m(u)\le t\}=(0,F_m(t)]\cap(0,1)$, an interval; so $q_m$ is Borel measurable on $(0,1)$ (rays generate). Let $\Omega_0=\bigcap_m\{U_m\in(0,1)\}$; by Step 2 and countable subadditivity (a consequence of additivity and monotonicity), $P(\Omega_0)=1$. Define $X_m=q_m(U_m)$ on $\Omega_0$ and $X_m=0$ on $\Omega\setminus\Omega_0$. Each $X_m$ is a random variable: with $C_{m,B}=q_m^{-1}(B)\cap(0,1)\in\mathcal{B}(\mathbb{R})$,

where the last term is present exactly when $0\in B$.

(Distribution.) For $t\in\mathbb{R}$, by the key equivalence and $P(\Omega_0)=1$,

using the distribution of $U_m$ from Step 2 (for $F_m(t)<1$ this is $F_m(t)$; for $F_m(t)=1$ it is $P(0<U_m\le1)=1$). Thus the probability measures $P_{X_m}$ and $\nu_m$ agree on the closed rays and on $\mathbb{R}$; the agreement class is a $\lambda$-system as before, containing the generating $\pi$-system, so $P_{X_m}=\nu_m$ by Dynkin's Pi-Lambda Theorem.

(Independence.) Fix distinct $m_1,\dots,m_p$ and Borel $B_1,\dots,B_p$. Each event $\{X_{m_a}\in B_a\}$ differs from $\{U_{m_a}\in C_{m_a,B_a}\}$ only within the $P$-null set $\Omega\setminus\Omega_0$, so replacing the former by the latter changes none of the probabilities below (monotonicity and additivity). By independence of $(U_k)$,

Hence $(X_m)_{m\in\mathbb{N}}$ is independent in the sense of Independence of Events and of Random Variables, and $X_m$ has distribution $\nu_m$ for every $m$. Taking all $\nu_m$ equal to a fixed $\nu$ recovers Existence of Independent and Identically Distributed Sequences, as noted in the statement. $\blacksquare$