A Functional Equation Whose Unknown is $\mathcal{P}([0,1])$ Valued

Abstract

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.

Appendix: Doob’s Decomposition of the RRU Process

This appendix provides a series of auxiliary results necessary to prove Propositions 3.1 and 3.2. We will refer to the notations introduced in Sect. 3. For n=1,2,…, let $\mathcal{A}_{n}=\sigma(\delta _{1},R_{X}(1),R_{Y}(1),\ldots,\delta_{n},R_{X}(n),R_{Y}(n) )$ and consider the filtration $\{\mathcal{A}_{n}\}$; then, given the initial urn composition $(x,y) \in{\mathbb{S}}$, the Doob’s semi-martingale decomposition of Z _n (x,y) is

where {M _n } is a zero mean martingale and the previsible process {A _n } is eventually increasing (decreasing), again by [11, Theorem 2]. We also denote by {〈M〉 _n } the bracket process associated with {M _n }, i.e. the previsible process obtained by the Doob’s decomposition of $M^{2}_{n}$.

We first provide some auxiliary inequalities. As a consequence of [2, Lemma 4.1], we can bound the increments ΔA _n of the Z _n -compensator process and the increments Δ〈M〉 _n of the bracket process associated with {M _n }. In fact, an easy computation gives

and

where

and

Now, [2, Lemma 4.2] with $m=\int_{0}^{\beta}k\mu (dk)=\int_{0}^{\beta}k\nu(dk)$ gives

(25)

By applying [2, Lemma 4.1] with $h(x,t) = (\frac{x}{x+t})^{2}$, B _D =[2β,∞), D=D _n , R=R _X (n+1) or R=R _Y (n+1) and ${\mathcal{A}}={\mathcal{A}}_{n}$, one obtains:

(26)

on the set {D _n ≥2β}. Since

if D ₀≥2β, and thus β+D _n ≥3β, (25) together with (26) yields

(27)

Lemma A.1

For all k=1,2,…,

(28)

If, in addition, D ₀≥2β then, for all k,n=1,2,… ,

(29)

when d≥c≥0 and b _k =c+D _n −β+mk.

Proof

Let η ^∗ be a random variable independent of ${\mathcal{A}}_{\infty}$ and let η ₁ be a random variable independent of $\sigma({\mathcal{A}}_{\infty},\eta^{*})$ and such that η ₁/β has distribution Binomial(1,m/β). Define ${\mathcal{A}}_{k+n^{-}}^{*}=\sigma(\eta^{*},{\mathcal{A}}_{k+n-1},\mathbb{I}(k+n))$; by [2, Lemma 4.1], if D>0 is ${\mathcal{A}}_{k+n^{-}}^{*}$-measurable and 0≤R≤β with ${\mathbb{E}}(R)=m$ is independent of ${\mathcal{A}}_{k+n^{-}}^{*}$, one obtains

and thus

(30)

Therefore, for c≥0, by applying (30) k times, we get

(31)

where η _k is independent of $\sigma({\mathcal{A}}_{\infty})$ and η _k /β has distribution Binomial(k,m/β). Equation (28) is now a consequence of [13, (21)]: if $\tilde{\eta_{k}}\sim\textrm{Binomial}(k,r)$ and l>0, then

Apply this to (31) with n=0, $\tilde{\eta_{k}}=\eta_{k}/\beta$, l=D ₀/β and r=m/β to obtain (28).

Equation (29) is a consequence of [13, (25)]: if $\tilde{\eta_{k}}\sim\textrm{Binomial}(k,r)$ and l>1, then

Apply this to (31) with $\tilde{\eta_{k}}=\eta_{k}/\beta$, l=D _n +c/β (which is greater than 2) and r=m/β to obtain

Jensen’s inequality yields ${\mathbb{E}}((d+D_{n+k})^{-1} |{\mathcal{A}}_{n} ) \geq(d+D_{n}+mk)^{-1}$, and thus

Since

we get (29):

 □

The following Lemmas A.2 and A.3 provide inequalities which control the previsible and the martingale part of the process Z _n , respectively; they require that the initial composition of the urn is sufficiently large.

Lemma A.2

If D ₀≥2β, then

Proof

Apply (29) with n=0, c=m, d=β. Equation (25) then reads

if b _k =km+D ₀−(β−m). Since A ₀=0,

where the last inequality is true because β−m≤β≤D ₀/2. □

Lemma A.3

Let D ₀≥2β. For all n≥0,

Proof

Since Z _n+k (1−Z _n+k )≤1/4, by (27), one gets

Apply (29) with c=0 and d=β, obtaining

if b _k =km+D _n −β. Thus

since 2(D _n −β)≥2(D ₀−β)≥D ₀. □