In one of my papers, I needed an upper bound on the fractional moments of the geometric distribution. The integer moments have a nice-looking upper bound involving factorials. In addition, Mathematica showed that the same upper bound holds for fractional moments once we use Gamma function instead of factorials. The problem was, I could not prove it. Worse, I could not find a proof after much looking on the Internet.
This post contains a proof that for any real , the th moment of a geometric random variable with success probability is at most .
Imagine that a particle is walking in a two-dimensional space, starting at the origin . At every time-step (or “epoch”) it takes a vertical step. At every step, the particle either moves up by , or down by . This walk is “unbiased” in the sense that the up/down steps are equiprobable.
In this post, we will discuss some natural questions about this “unbiased” or “simple” random walk. For example, how long will it take for the particle to return to zero? What is the probability that it will ever reach +1? When will it touch for the first time? Contents of this post are a summary of the Chapter “Random Walks” from the awesome “Introduction to Probability” (Volume I) by William Feller. Also, this is an excellent reference.
Edit: the Biased Case. The biased walk follows
The lazy walk follows
In the exposition, we’ll first state the unbiased case and when appropriate, follow it up with the biased case.
[Contents of this post are based on an ongoing discussion with Alex Russell and Aggelos Kiayias. It contains potentially unpublished material.]
In a proof-of-stake blockchain protocol such as Ouroboros, at most half of the users are dishonest. While an honest user always extends the longest available blockchain, the dishonest users try to fool him into extending a manipulated blockchain. Here, the user who is allowed to issue a block at any time-slot is called the “slot leader.” As it happens, a number of future slot leaders are computed in advance using the random values present in the blocks. Although counterintuitive, such a scheme ensures that if the adversary does not control more than half the users now, it is very unlikely that he cannot control more than half the slot leaders. The time-slots are divided into “epochs” of length .
In a blockchain protocol such as Bitcoin, the users see the world as a sequence of states. A simple yet functional view of this world, for the purpose of analysis, is a Boolean string of zeros and ones, where each bit is independently biased towards favoring the “bad guys.”
A bad guy is activated when for some . He may try to present the good guys with a conflicting view of the world, such as presenting multiple candidate blockchains of equal length. This view is called a “fork”. A string that allows the bad guy to fork (with nonnegligible probability) is called a “forkable string”. Naturally, we would like to show that forkable strings are rare: that the manipulative power of the bad guys over the good guys is negligible.
Claim (, Bound 2). Suppose is a Boolean string, with every bit independently set to with probability for some . The probability that is forkable is at most .
In this post, we present a commentary on the proof that forkable strings are rare. I like the proof because it uses simple facts about random walks, generating functions, and stochastic domination to bound an apparently difficult random process.
Consider a random walk in the two-dimensional discrete space, where the horizontal direction is indexed by nonnegative time steps and the vertical direction is indexed by integers.
The particle is at its initial position at time . At every time step, it independently takes a step up or down: up with probability and down with probability . If the walk is called symmetric. Let be constant, . If the walk reaches either or , it stops. Define the two quantities below.
It just so happens that given enough time, the walk either ruins or escapes. We set the initial conditions as .
The Gambler’s Ruin Problem: What is the probability that a walk, starting at some , eventually reaches the origin?