I think it is impossible to answer this question without carrying out an extensive genetic sampling from the current US population. I am not aware that such data exist, but this is what one should be looking for.
Making inferences from demographics/migration data (as suggested in some of the comments) may be very misleading: genes (more precisely alleles) do not propagate through the population uniformly/proportionally even in absence of selection - due to genetic drift some Y-chromosomes may be overrepresented, while many have gone completely extinct. When selection is in play, the consequences can be even more spectacular - like 16 million descendants of Genghis Khan (see the links to research papers in this article).
To perform a simple estimate we could define as
nt the number of individuals that immigrated in year
t from the beginning of the period of interest, so that
n0 is the number of individuals that were already present in the beginning of this period. We assume that the period is
T years long, and that the current population is
N. The important parameter is the yearly increase of the population:
b, d - rates of births and deaths), so that the number of the offspring produced by the immigrats from year
Nt = nt * s^(T-t)
s can be determined by summing all the
Nt, which gives us the total current population in the US
N0 + N1 + N2 + ... + NT = n0*s^T + n1*s^(T-1) + n2*s^(T-2) + ... + nT=N.
This equation can be solved numerically or graphically.
The fraction of the population descending from year
t is then
Difficulties with the demographic approach
Stochasticity of birth and deaths
The OP suggest using Y-chromosome (only paternal descent) or mitochondrial genome (only maternal descent). This could be very useful, if we actually had the genetic data sampled from the current population, which are likely unavailable. It is necessary however to point out the complication which immediately arises in such a picture: the births and deaths are a stochastic process: some fathers/mothers left no children behind, whereas others have left multiple offspring. These fluctuations grow with the rate faster than
s, and likely make the inference above meaningless (i.e., the standard deviations of
Nt are of the order of
If we do not limit ourselves to only paternal or only maternaldescent then we have to account for another difficulty, as pointed in the comments by @jamesqf: since each person has a father and a mother, one has two anscestors in the previous generation, four ancestors two generations back, eight ancestors three generations back and so on. Thus, most persons cannot be said do descend from a specific immigration year - they likely have ancestors in many years. On thus could ask a question somewhat different from the reasoning suggested in the OP: what percentage of the current population have an ancestor that came with a certain immigration wave?
The estimate actually works for the US population and not too small immigrant waves (e.g., Mayflower with its 100 passangers may be abit problematics). See the question and answer in the biology forum.