Asymptotic probability distribution of distances between local extrema of error terms of a moving average process

Consider error terms x(i) of a moving average process MA(q), where x(i)=e(i) + e(i-1)+...+e(i-q) and e(i) - independent identically distributed (i.i.d.) random variables. We recognize a term x(i) as a local maximum if the following condition holds true: x(i-1) < x(i) > x(i+1). If the local maximum x(i) is followed by the next local maxiumum x(k), then d=k-i is the distance between local maxima. The distances d(j) themselves are random vriables. In this paper we study the probability distribution of distances d(j). Particularly, we show that for any q>0 mean distance E[d(j)]=4 and asymptotically the variance is also equal to 4.