Is the ratio of two sums normally distributed? I've encountered this problem and I'm not sure whether my logic is correct.  Suppose I have a random sample of customer spending and I want to estimate the market share of a given store in a given quarter (which I define to be the proportion of sales made in that store against the sum of sales in all other competing stores in the locality).
I calculate this statistic by taking the ratio of total spend in the store of interest against the sum of total spend.  In equation form the statistic of interest is:
$$
Market \ share = \frac{\sum_{c} X_c}{\sum_c Y_c}
$$
where $c$ is a customer in the sample, $X_c$ is total spending by customer c on store X and $Y_c$ is total spending by customer c on all stores of the same type as store $X$.
Suppose I want to estimate a confidence interval for this estimate.  I know by the Central Limit Theorem that the total sum spent, $\sum_{c} X_c$ and $\sum_c Y_c$, are both going to be normally distributed asymptotically.
However, since the market share is the ratio of two sums my intuition suggests that this statistic will not be asymptotically normally distributed (as the ratio of two normal distribution is not normally distributed in general).  Due to my hesitance about the asymptotic distribution of this statistic I decided to construct the confidence interval by bootstrapping instead.
My questions on this issue would be, am I ignorant of some nice asymptotic result that I could use and is my logic correct towards why I should bootstrap this problem?

Experiment. The ratio is Cauchy, as noted, if numerator and denominator are both normal distributions centered at zero (not illustrated here).
However, the ratio is also often
nowhere near normal if numerator and denominator are both centered several
standard deviations above zero. Using R:
set.seed(1234)
x1 = rnorm(100, 50, 7)
x2 = rnorm(100, 70, 8)
ratio = x1/x2
shapiro.test(ratio)

        Shapiro-Wilk normality test

data:  ratio
W = 0.94094, p-value = 0.0002201

qqnorm(ratio); qqline(ratio, col="blue")


hist(ratio, prob=T, col="skyblue2")


