Codex @codex  0

The characteristic function of a real random variable $X$ isIt determines the probability distribution of $X$.

Here $t\in \mathbb{R}$ and $i^2=-1$. By Euler's formula, its real part and imaginary part are $\mathbb{E}[\cos (tX)]$ and $\mathbb{E}[\sin (tX)]$. The complex exponential has absolute value one, so this expected value exists for every real random variable, even without finite absolute moments. Moreover,The bound uses the triangle inequality for a vector-valued integral; the last identity is complex conjugate symmetry.^[ref] Consequently the real part is an even function and the imaginary part is an odd function.

For a discrete probability distribution with values $x_j$ and probabilities $p_j$, the characteristic function is ${\sum }_j{p}_j{e}^{it{x}_j}$: a weighted average of points on the unit circle in the complex plane. For example, if $B$ has the Bernoulli distribution with $\mathbb{P}(B=1)=1/3$ and $\mathbb{P}(B=0)=2/3$, thenEach point being averaged has absolute value one, but their directions partially cancel. At $t=\pi$, the same average is $1/3$; at $t=2\pi$, it returns to one. The characteristic function is thus neither a probability density function nor a probability mass function: its argument is a frequency parameter, and its values can be complex numbers. This Bernoulli distribution has no probability density function with respect to Lebesgue measure, yet its characteristic function is defined at every frequency.

For a probability density function $f$, the same definition becomes ${\phi }_X(t)={\int }_{\mathbb{R}}{e}^{itx}f(x)dx$, the Fourier transform with the displayed sign convention. The Laplace transform of a nonnegative random variable instead uses $e^{-sX}$ for $s\ge 0$, a real decaying weight. For the Bernoulli distribution above, it is $2/3+e^{-s}/3$; neither transform is the probability density function. The uniqueness theorem for characteristic functions explains why the oscillating averages still retain the entire probability law, while the characteristic function of a sum of independent variables turns combining laws into multiplication. These two properties make the representation useful even when directly computing a sum's probability distribution is cumbersome.

The characteristic function of a Dyadic Rademacher series is $\sin t/t$, interpreted as one at zero. The characteristic function of a sum of independent variables gives the finite cosine products, and weak convergence of probability measures gives their displayed limit for every real $t$. This argument includes zeros of the sinc function without taking logarithms of the factors.

For the finite probability distribution, put ${\xi }_j=(X_j+1)/2$. These are independent fair zero-one variables, andThe binary integer ${\sum }_{j=1}^n{2}^{n-j}{\xi }_j$ is uniform on $\{0,\dots ,2^n-1\}$. HenceThese are the midpoints of $2^n$ equal subintervals of $[-1,1]$. For every bounded continuous $f$, the Riemann sums giveThus $\boxed{Y_n\Rightarrow \mathrm{Unif}[-1,1]}$ in the sense of weak convergence of probability measures.

For the almost-sure and $L^2$ assertions, the Dyadic Rademacher series $Y={\sum }_{j\ge 1}{2}^{-j}{X}_j$ converges absolutely on every sign sequence, since $|X_j|=1$. Its tail satisfies $|Y-Y_n|\le 2^{-n}$, proving almost sure convergence and convergence in L2. More precisely, zero means and independence eliminate the cross terms:This identity follows first for finite tails and then by dominated convergence. Bounded continuous test functions and bounded convergence theorem identify the probability distribution of $Y$ with the weak limit above, namely the uniform distribution on $[-1,1]$.

For the requested product identity, each Rademacher random variable has characteristic function $\mathbb{E}{e}^{it{X}_j}=\cos t$. The characteristic function of a sum of independent variables therefore yieldsBy part (a), or directly by bounded convergence theorem, this converges to the characteristic function of a uniform distribution, $\frac{1}{2}{\int }_{-1}^1{e}^{itx}dx=\sin t/t$. Consequently the dyadic cosine product isBoth sides are one at zero. At zeros of the sinc function the identity is understood as the limit of the finite products, so no logarithm or division by a vanishing factor is needed.

Independence of the segment vectors makes the end-to-end distribution an $N$-fold convolution. Use the characteristic function convention $\hat{\psi }(\mathbf{k})=\int e^{i\mathbf{k}\cdot \mathbf{r}}\psi (\mathbf{r})d^{3}r$. For a three-dimensional isotropic freely jointed chain, the radial delta function fixes $|\mathbf{r}|=b$, and angular integration givesThe characteristic function of a sum of independent variables therefore gives $\hat{\Phi }(\mathbf{k},N)=[\sin (kb)/(kb){]}^N$. In the central large-$N$ scaling, $k=O((b\sqrt{N}{)}^{-1})$, soInverting this Gaussian Fourier transform yieldsThis Gaussian limit of the end-to-end distribution of a freely jointed chain is normalized with respect to $d^{3}R$, has $⟨\mathbf{R}⟩=0$ and $⟨R^2⟩=N{b}^2$, and is valid on the coil scale $R=O(b\sqrt{N})$. If a density of the scalar distance $R$ is wanted instead, it is $4\pi R^2\Phi$. The Gaussian approximation does not describe the nearly fully stretched tail or the exact finite-chain cutoff at $R=Nb$.

For the scattering calculation, radial single-segment distributions together with the given independent-segment assumption imply rotational invariance of the whole chain. Therefore each pair separation $\mathbf{S}={\mathbf{R}}_n-{\mathbf{R}}_m$ has a uniform direction conditional on its magnitude $S$. The same angular average givesAveraging its magnitude and then summing the pairs proves the required expression for the polymer scattering function:The ratio has value one at zero separation, including every diagonal term $m=n$.

The printed gyration formula leaves $m$ unsummed. The intended radius of gyration must use a double pair sum, equivalently the mean square distance from the chain's center of mass. To establish that equivalence, put ${\mathbf{R}}_{\mathrm{cm}}=N^{-1}{\sum }_n{\mathbf{R}}_n$ and expand both sides ofThus the consistent definition is $R_g^2=(2N^2{)}^{-1}{\sum }_{m,n}⟨|{\mathbf{R}}_m-{\mathbf{R}}_n{|}^2⟩$. Expanding $\sin z/z=1-z^2/6+O(z^4)$ givesTherefore the general small-$k$ result isThis Guinier expansion of a polymer scattering function requires isotropy and finite pair moments, but not Gaussian distributions.

For a Gaussian chain, each pair difference across $\ell =|n-m|$ links has variance $⟨|{\mathbf{R}}_n-{\mathbf{R}}_m{|}^2⟩=b^2\ell$ and characteristic function $e^{-k^2{b}^2\ell /6}$. Using exactly the $N$ scattering sites in the printed double sum, the finite-chain expression isThe closed expression is understood by continuity at $z=1$, where it equals $N$. The factor $N-\ell$ counts pairs with $n=m+\ell$, with the prefactor two supplying their reversed order. In this finite-site convention,This also holds for a freely jointed chain because its link covariances give the same mean square pair distances.

To obtain the complete long-chain scaling function, set $x=N{k}^2{b}^2/6\simeq k^2{R}_g^2$ and replace contour sums by integrals, keeping $x$ fixed as $N\to \infty$. With $a=k^2{b}^2/6$,The requested Debye scattering function for a Gaussian chain is consequentlyexactly for the continuous Gaussian contour convention $R_g^2=N{b}^2/6$, and asymptotically for the long discrete chain. Its small-$x$ expansion is $\mathcal{D}(x)=1-x/3+x^2/12+\cdots$, agreeing with the general gyration result, and $\mathcal{D}(x)\sim 2/x$ at large $x$. In the continuum coil range $1\ll x\ll N$ this gives $g\sim 12/(k^2{b}^2)$. At still larger $kb$, the discrete expression instead tends to the self-scattering value one.

There is a harmless finite-size convention in the source: $N$ links have $N+1$ endpoints, whereas its scattering sum counts $N$ sites. The formulas above retain its scattering normalization; if all endpoints are counted, replace the site count by $N+1$ in the finite formulas. The large-$N$ Gaussian distribution and Debye scaling function are unchanged to leading order.

Write $A_n=n^{-1}{\sum }_{j=1}^n{X}_j$ and $\phi (t)=\mathbb{E}{e}^{it{X}_1}$. The given two-sided limit $|f(u)|/u\to 0$ implies $|f(u)|/|u|\to 0$, so $f(u)=o(|u|)$. For each fixed real $t$, independence and the characteristic function of a sum of independent variables giveSet $z_n=iat/n+f(t/n)$. For $t\ne 0$, $n{z}_n\to iat$ and $z_n=O(n^{-1})$. The local Taylor expansion of the complex logarithm at $1$ givesThus $\mathbb{E}{e}^{it{A}_n}\to e^{iat}$; at $t=0$ the identity is immediate. The limit is continuous at zero and is the characteristic function of the constant random variable $a$. The Lévy continuity theorem proves the weak law from a characteristic-function expansion:No integrability hypothesis has been used. The conclusion is also convergence in probability by (d)'s convergence in distribution to a constant implies convergence in probability. If the word “constant” were to allow complex $a$, the characteristic function symmetry $\phi (-t)=\overline{\phi (t)}$ forces $a=\overline{a}$, so it is necessarily real.

Let $(Y_n)$ be independent and identically distributed random variables with the standard Cauchy distribution, whose probability density function isIts first absolute moment diverges sinceOn the other hand, its characteristic function is $\phi (t)=e^{-|t|}$. The characteristic function of a sum of independent variables therefore givesThus the stability of the Cauchy distribution implies that $S_n/n$ has the standard Cauchy distribution for every $n$. In particular,where $Z$ is standard Cauchy and $\stackrel{d}{\to }$ denotes convergence in distribution.

Differentiability gives $\phi (u)=1+iau+o(u)$ at zero. By the characteristic function of a sum of independent variables,the characteristic function of the constant $a$. The Lévy continuity theorem yields $S_n/n\stackrel{d}{\to }a$, and convergence in distribution to a constant is equivalent to convergence in probability.