# Application of law of large numbers for triangular arrays

Consider a sequence of i.i.d. random variables $$\{\xi_j\}_{j=1}^J$$.

Consider a random variable $$X$$ with support $$\mathbb{R}$$.

Consider the map $$f: \mathbb{\mathbb{R}}\rightarrow \mathbb{R}^J$$. Let $$f_j(X)$$ be the $$j$$-th element of the $$J\times 1$$ random vector $$f(X)$$.

Assume that $$E(\xi_j| X)=0$$ $$\forall j=1,...,J$$.

I want to show that $$\frac{1}{J} \sum_{j=1}^J f_j(X) \xi_j\rightarrow_p0 \text{ as J\rightarrow \infty}$$

The book I'm reading claims that this holds because:

• The random variables in the sequence $$\{f_j(X) \xi_j\}_{j=1}^J$$ are i.i.d. conditional on $$(f_j(X) \text{ }\forall j=1,...,J)$$.

• Hence, under the law of large numbers for triangular arrays, $$\frac{1}{J} \sum_{j=1}^J f_j(X) \xi_j\rightarrow_p0$$ as $$J\rightarrow \infty$$

I'm struggling to understand this proof.

The law of large number for triangular arrays states:
Consider the triangular sequence $$\{(Y_{J,j})_{j=1}^J\}_{J \in \mathbb{N}}$$. Assume $$Y_{J,1},\ldots,Y_{J,J}$$ are i.i.d random variables with mean $$\mu_J$$. Then, under some conditions [?], it holds that $${1 \over J}\sum_{j=1}^J Y_{J,j}-\mu_J \to_p 0 \text{ as J\rightarrow \infty}$$ (see here for example).

But what is $$Y_{J,j}$$ in my example? Could you help me to clarify?

• Does the book give any conditions that $f$ has to satisfy? After all, if $f_j(x)=2^jx$ then $\frac{1}J\sum_{j=1}^Jf_j(X)\eta_j$ might not converge to zero. Jun 9, 2019 at 12:06

My interpretation:

First of all, $$f$$ should really be $$f^J$$ since the domain of $$f$$ is $$\mathbb{R}^J$$ so different $$J$$'s $$\implies$$ different $$f$$'s.

Then the thing you want to prove becomes (edit in red):

$$\frac{1}{J} \sum_{j=1}^J f^\color{red}{J}_j(X) \xi_j\rightarrow_p0 \text{ as J\rightarrow \infty}$$

Then comparing to your theorem for triangular array, we identify $$Y_{J,j} \equiv f^J_j(X) \xi_j$$, and note that:

$$\mu_J = E[Y_{J,j}] = E[f^J_j(X) \xi_j] = E_X [ E_\xi [\xi_j | f^J_j(X)]] = E_X [ 0] = 0$$

However, as pointed out by the comment of @AngelaRichardson, you might be missing some condition on $$f$$ (and/or on $$Y$$). E.g. consider:

$$f^J_j(X) = \begin{cases} JX, & j = J\\ 0, & j < J \end{cases}$$

Then:

$$\frac{1}{J} \sum_{j=1}^J f^J_j(X) \xi_j = \frac{1}{J} (JX) \xi_j = X \xi_j \not\to_p 0$$

Even if you additionally impose the condition that $$f_j^J = f_j^K$$ for all $$j,J,K$$ in range, this would rule out the above example but the example by Angela that $$f_j(X) = 2^j X$$ is still likely to lead to $$\not\to_p 0$$.

• Thanks. Can you provide some sufficient conditions for the triangular law of large numbers to hold (e.g., finiteness and existence of some moments, etc....)? Maybe the book considers them implicit. Thanks.
– Star
Jun 10, 2019 at 8:19
• Also, it is still unclear to me how we get i.i.d.ness of $\{f_j^J(X)\xi_j\}_{j=1}^J$ from the assumptions.
– Star
Jun 10, 2019 at 11:00
• I personally have not heard of the triangular law until this post. I was simply trying to fit the first part of what you said into the second part of what you said. :) However, you have a good point... I'd think $\{f^J_j(X) \xi_j\}_{j=1}^J$ are NOT i.i.d. because the components $f_j$ can certainly correlate. Even your book only claimed they are i.i.d. when conditioned on ${f^J_j(X)}_{j=1}^J$, but even that seems false: they might be independent when conditioned, but how can they be identically distributed? The $\xi$'s are i.d. but when scaled by $f_j$ they shouldn't be i.d. Jun 10, 2019 at 15:39