I need to calculate the average of the following quantity:

\begin{equation} S_n=\prod_{i=1}^nS(X_i) \tag{1} \label{eq:1} \end{equation}

with $S(X_i):=o_{X_i}b_{X_i}$, where each $X_i\in \mathcal{X}=\{1,2,\dots,m\}$ is drawn i.i.d. according to a probability mass function $\mathbf{p}=\{p_1,p_2,\dots,p_m\}$. The $o_X\in\{o_1,o_2\dots,o_m\}$ are fixed real numbers with each $o_j \geq 0$ and the $b_X\in \mathbf{b}=\{b_1,b_2\dots,b_m\}$ are also fixed real numbers with $b_j\geq 0$ and $\sum_{j=1}^mb_j=1$. A special case could be $\mathbf{b}=\mathbf{p}$.

The average I need can be written as follows, using equation (\ref{eq:1}):

\begin{equation} \bar{S_n}=\frac{1}{n} \sum_{k=1}^n S_k=\frac{1}{n} \sum_{k=1}^n \prod_{i=1}^k S(X_i) \tag{2} \end{equation}

Which can be rewritten as:

\begin{equation} \bar{S_n}=\frac{1}{n} \{S(X_1)+S(X_1)S(X_2)+\cdots + S(X_1)\cdots S(X_n)\} \tag{3} \end{equation}

Question. Is there a way to use the weak law of large numbers in order to write this average in terms of the probability mass function $\mathbf{p}$?

  • $\begingroup$ If I understand well the question, you would like to know the limit in probability of the sequence $\left(\bar{S_n}\right)_{n\geqslant 1}$, right? $\endgroup$ – Davide Giraudo May 26 '16 at 16:42
  • $\begingroup$ @DavideGiraudo That's right. $\endgroup$ – Ana S. H. May 27 '16 at 21:32
  • $\begingroup$ @DavideGiraudo The asymptotic behavior for large $n$ would be useful too. $\endgroup$ – Ana S. H. May 27 '16 at 21:44

It will depend on the parameters, as one could expect. For example, if $p_i=b_i=1/m$ and $o_i=1/b_i^2$, then $S(X_i)=m$ and the sequence $\left(\bar{S_n}\right)_{n\geqslant 1}$ goes to infinity.

However, if $\mathbb E\left[S\left(X_1\right)^2\right]\lt 1$, then $\bar{S_n}\to 0$ in probability (actually even in $\mathbb L^2$). To see this, expand the square and use independence of the sequence $\left(S(X_i)\right)_{i\geqslant 1}$.


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