Design and Implementation of a Homomorphic ... - Researcher

H/2m each and 0 with probability 1 − H/m. Let f(X) = ∑ m−1
i=0 f iX i . Then for fixed r and
H, m → ∞, we have
E[|f(τ)| 2r ] ∼ r!H r .
In particular, for H ≥ 2r 2 , we have
E[|f(τ)| 2r ]
∣ r!H r − 1
∣ ≤ 2r2
H + 2r+1 r 2
m .
Before proving Theorem 1, we introduce some notation and prove some technical results that
will be useful.
Recall the “falling factorial” notation: for integers n, k with 0 ≤ k ≤ n, we define n k =
∏ k−1
j=0
(n − j).
Lemma 1. For n ≥ k 2 > 0, we have n k − n k ≤ k 2 n k−1 .
Proof. We have
n k ≥ (n − k) k = n k −
( ( ( k k k
kn
1)
k−1 + k
2)
2 n k−2 − k
3)
3 n k−3 + − · · · .
The lemma follows by verifying that when n ≥ k 2 , in the above binomial expansion, the sum of
every consecutive positive/negative pair of terms in non-negative.
Lemma 2. For n ≥ 2k 2 > 0, we have n k ≤ 2n k .
Proof. This follows immediately from the previous lemma.
Next, we recall the notion of the Stirling number of the second kind, which is the number of
ways to partition a set of l objects into k non-empty subsets, and is denoted { l
k}
. We use the
following standard result:
l∑
{ l
n
k}
k = n l . (1)
k=1
Finally, we define M 2n to be the number of perfect matchings in the complete graph on 2n
vertices, and M n,n to be the number of perfect matchings on the complete bipartite graph on two
sets of n vertices. The following facts are easy to establish:
and
M n,n = n! (2)
M 2n ≤ 2 n n!. (3)
We now turn to the proof of the theorem. We have
f(τ) 2r = f(τ) r f(¯τ) r =
∑
f i1 · · · f i2r · τ i1 · · · τ ir · τ −i r+1
· · · τ −i 2r
.
i 1 ,...,i 2r
We will extend the usual notion of expected values to complex-valued random variables: if U and
V are real-valued random variables, then E[U + V i] = E[U] + E[V ]i. The usual rules for sums and
products of expectations work equally well. By linearity of expectation, we have
E[f(τ) 2r ] =
∑
E[f i1 · · · f i2r ] · τ i1 · · · τ ir · τ −i r+1
· · · τ −i 2r
. (4)
i 1 ,...,i 2r
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