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A concept indicating that the estimator is unbiased in the limit (cf. [[Unbiased estimator|Unbiased estimator]]). Let  $  X _ {1} , X _ {2} \dots $
be a sequence of random variables on a probability space  $  ( \Omega , S, P ) $, 
where  $  P $
is one of the probability measures in a family  $  {\mathcal P} $. 
Let a function  $  g(P) $
be given on the family  $  {\mathcal P} $, 
and let there be a sequence of  $  S $-
measurable functions  $  T _ {n} ( X _ {1} \dots X _ {n} ) $, 
$  n = 1, 2 \dots $
the mathematical expectations of which,  $  {\mathsf E} _ {P} T _ {n} ( X _ {1} \dots X _ {n} ) $, 
are given. Then, if, as  $  n \rightarrow \infty $,

$$ 
{\mathsf E} _ {P} T _ {n} ( X _ {1} \dots X _ {n} )  \rightarrow \ 
g (P),\  P \in {\mathcal P} ,
$$

one says that  $  T _ {n} $
is a function which is asymptotically unbiased for the function  $  g $. 
If one calls  $  X _ {1} , X _ {2} \dots $"
observations"  and  $  T _ {n} $"
estimators" , one obtains the definition of an asymptotically-unbiased estimator. In the simplest case of unlimited repeated sampling from a population, the distribution of which depends on a one-dimensional parameter  $  \theta \in \Theta $, 
an asymptotically-unbiased estimator  $  T _ {n} $
for  $  g ( \theta ) $, 
constructed with respect to the sample size  $  n $, 
satisfies the condition

$$ 
{\mathsf E} _  \theta  T _ {n} ( X _ {1} \dots X _ {n} )
 \rightarrow  g ( \theta )
$$

for any  $  \theta \in \Theta $, 
as  $  n \rightarrow \infty $.