Probability distribution of the distance after $n$ random jumps in a plane

analysisprobability distributionsprobability theoryrandom walkreference-request

I want to know if there is an analytical way to approximate the distribution of a random variable defined by

$$Y_n:=\left|\sum_{k=1}^n e^{i X_k}\right|$$

where the $X_k\sim U[-\pi,\pi]$ are i.i.d. I did some computer simulations, however Im trying to see if there is some analytic machinery to, at least, approximate the distribution of $Y_n$.

I know how to write analytically $E[Y_n]$ for $n$ (indeed I have the estimate $E[Y_n]\le\sqrt n$, via Jensen's inequality because $E[Y_n^2]=n$), and I could write explicitly $F_{Y_2}$, however for $n>2$ I dont know how to proceed (or how to approximate).

Also I know how to explicitly write the iterated integrals for the computation of the distribution of $\sum_{k=1}^n e^{i X_k}$, however I dont know any approach to compute the distribution of it absolute value.

Some help will be appreciated or, if someone knows, some paper or book where to dig about similar questions.

EDIT: also note that $Y_n=|Y_{n-1}+e^{i X_n}|$, so it seems possible to approximate the distribution of $Y_n$ using some kind of recursion.

Also it is easy to check that

$$Y_n=\sqrt{n+2\sum_{1\le j< k\le n}\cos(X_j-X_k)}$$

However this last expression at first glance doesn't seems useful for an analytic (approximation) to it distribution.

EDIT 2: adding some histograms that approximate the densities

enter image description here

For $n\le 5$ the estimated densities are strange (not of bell form), however for $n=2$ the density approaches the theoretical density, that is

$$f_{Y_2}(x)=\frac1{\pi\sqrt{1-(x/2)^2}}\chi_{[0,2]}(x)$$

(Note the vertical asymptote at $x=2$ in $f_{Y_2}$. Anyway the empirical approximation tends to it). If someone is interested this was the code, written in Julia, that I had used for the simulations:

# Sum of random jumps in the plane
function rsum(n::Int)
    p = 1.0
    for j in 2:n
        p += exp(2pi * im * rand())
    end
    return p
end

# Distance after n random jumps
function rd(n::Int)
    abs(rsum(n))
end

 # Data array to build an histogram
 function sim(n::Int, m::Int = 22)
    datos = zeros(2^m)
    for i in 1:2^m
        datos[i] = rd(n)
    end
    return datos
 end

# Plotting densities
using StatsPlots, Statistics
function dd(n::Int,m::Int=22)
    x = sim(n,m)
    density!(x, w = 2,
        xlabel = "Distance",
        label = "Estimated density for $n jumps",
        fill = (0, 0.1, :orange))
    vline!([mean(x)],
        label = "Estimated mean for $n jumps: $(round(mean(x),digits=2))",
        line = :dash)
end

Best Answer

Since we'll be working with both real and imaginary parts, it's convenient to consider them as components of a vector: $$ V_k = [\cos(X_k), \sin(X_k)] $$ Note that these have mean $0$ and covariance matrix $$ \Sigma = \pmatrix{1/2 & 0\cr 0 & 1/2\cr}$$ By the multivariate version of the Central Limit Theorem, if $S_n = \sum_{k=1}^n V_k$, $ S_n/\sqrt{n} $ tends in distribution to a bivariate normal random variable with mean $0$ and covariance matrix $\Sigma$. In particular, the distribution of $|S_n|/\sqrt{n}$ approaches that of $\sqrt{(Z_1^2 + Z_2^2)/2}$ where $Z_1$ and $Z_2$ are independent standard normal random variables. It can be shown (by using polar coordinates) that this has PDF $2 r e^{-r^2}$ for $r > 0$.

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