A New Proof of Central Limit Theorem for i.i.d. Random Variables

Proposition 1. Suppose $$ is a sequence of i.i.d. random variables. Then

• (i)

  for each f ∈ C(ℝ), if E[|f(X₁)|] < ∞, then ∀i, j ∈ ℕ,

  $$ ()

• (ii)

  ∀i ∈ ℕ; for each f ∈ C(ℝⁱ⁺¹), if E[|f(Y, X_i+1)|] < ∞, then

  $$ ()
  where Y : = (X₁, …, X_i).

Proposition 2. Suppose X is a normally distributed random variable with E[X] = 0 and E[X²] = σ² > 0, denoted by N(0, σ²). Then if $$ and Y is independent of X, we have, for each f ∈ C_b,Lip(ℝ),

We will show that a normally distributed random variable X with E[X] = 0 and E[X²] = σ² > 0 is characterized by the following PDE defined on [0, ∞) × ℝ:

with Cauchy condition u(0, x) = f(x). Equation (7) is called the heat equation.

Definition 3. A real-valued continuous function u ∈ C([0, ∞) × ℝ) is called a viscosity subsolution (resp., supersolution) for (7), if for each function $$ and for each minimum (resp., maximum) point (t, x)∈[0, ∞) × ℝ of v − u, we have

u is called a viscosity solution for (7) if it is both a viscosity subsolution and a viscosity supersolution.

Remark 4. For more basic definitions, results, and related literature on viscosity solutions of PDEs, the readers can refer to Crandall et al. [12].

Lemma 5. Let  X  be an N(0, σ²) distributed random variable. For each f ∈ C_b,Lip(ℝ), we define a function

Then we have

We also have the estimates: for each T > 0, there exists a constant C > 0 such that, for all t, s ∈ [0, T] and x, y ∈ ℝ,

Moreover, u is the unique viscosity solution, continuous in the sense of (11) and (12), of (7) with Cauchy condition u(0, x) = f(x).

Proof. Since

we then have (11). Let  Y  be independent of X such that $$. By Propositions 1 and 2, we have

It follows from this and (11) that

which implies (12).

Now, for a fixed point (t, x)∈(0, ∞) × ℝ, let $$ satisfy v ≥ u and v(t, x) = u(t, x). By (10), we have, for δ ∈ (0, t),

where $$ is a positive constant, and then, we have

Hence, u is a viscosity subsolution for (7). Similarly, we can prove that u is a viscosity supersolution for (7). The proof of Lemma 5 is completed.

Theorem 6. Let $$ be a sequence of i.i.d. random variables. We further assume that

Denote $$. Then

In order to prove Theorem 6, we need the following lemma.

Lemma 7. Under the assumptions of Theorem 6, we have

for any f ∈ C_b,Lip(ℝ), where X is N(0, σ²).

Proof. The main approach of the following proof derives from Peng [10]. For a small but fixed h > 0, let v be the unique viscosity solution of

By Lemma 5,

Particularly,

Since (21) is a uniformly parabolic PDE, thus by the interior regularity of v (see Wang [13]), we have

We set δ∶ = 1/n and S₀∶ = 0. Then

By Taylor’s expansion,

Thus

We now prove that

Indeed, for the 3rd term of $$, by Proposition 1,

For the second term of $$, by Proposition 1, we have

Thus combining the above two equalities with

we have

Thus, (27) can be rewritten as

But since both ∂_tv and $$ are uniformly α-hölder continuous in x and α/2-hölder continuous in t on [0,1] × ℝ, we then have

Thus

where C is a positive constant. As n → ∞, we have

On the other hand, for each t, t^′ ∈ [0,1 + h] and x ∈ ℝ,

Thus

and by (23)

It follows from (23), (36), (38), and (39) that

Since h can be arbitrarily small, we have

Proof of Theorem 6. For notional simplification, write

Let ε be any positive number, and take δ small enough such that $$. Construct two functions f, g such that

Then

and for each n,

Obviously, f and g ∈ C_{b, Lip} (ℝ). By Lemma 7, we have

So

Hence

Since this is true for every ε, we have