Distribution, type of

A set of probability distributions of random variables, each obtainable from the others by means of linear transformations. An exact definition in the one-dimensional case is as follows: Probability distributions of random variables $X_1$ and $X_2$ are said to have the same type if there are constants $A,B>0$ such that the distributions of $X_2$ and $BX_1+A$ coincide. The corresponding distribution functions are then connected by the relation

$$F_2(x)=F_1\left(\frac{x-A}{B}\right)=F_1(bx+a),$$

where $b=1/B$ and $a=-A/B$.

Thus the set of distribution functions decomposes into mutually disjoint types. For example, all normal distributions form one type, and all uniform distributions form another.

The concept of type is widely used in limit theorems of probability theory. The distributions of the sums $S_n$ of independent random variables often "unboundedly diverge" as $n\to\infty$, and the convergence to a limit distribution (such as a normal distribution) is possible only after linear "normalization" , i.e. for sums $(S_n-a_n)/b_n$, where $a_n,b_n>0$ are constants and $b_n\to\infty$ as $n\to\infty$. In addition, if for random variables $X_n$ the distributions of $(X_n-a_n)/b_n$ and $(X_n-a_n')/b_n'$ converge to non-degenerate limit distributions, then these must be of the same type. Therefore it is possible to give the following definition of convergence of types (A.Ya. Khinchin, 1938). Let $T(F)$ be the type of the distribution function $F$ (the degenerate type is excluded from this discussion, that is, the type of degenerate distributions). One says that a sequence of types $T_n$ converges to a type $T$ if there is a sequence of distribution functions $F_n\in T_n$ that is (weakly) convergent to a distribution function $F\in T$. The set of types topologized in this way is a non-regular Hausdorff space and is thus non-metrizable (W. Doeblin, 1939).

Now, let $S_n$ be sums of independent identically-distributed random variables with corresponding distribution functions $F_n$. Then the class of limit types of the $T(F_n)$ coincides with the class of all stable types, i.e. types $T$ such that whenever $F_1$ and $F_2$ belong to $T$, so does the convolution of $F_1$ and $F_2$ (in other words, the sum of two independent random variables with distributions of type $T$ again has type $T$; see Stable distribution).

The concept of a type of distribution can be extended to the multi-dimensional case. However, this extension is ambiguous. For any subgroup $G$ of the full matrix group one obtains a corresponding concept of a type of distribution. Random vectors $X_1$ and $X_2$ with values from $\mathbf R^n$ are said to have the same $G$-type if there is a transformation $g\in G$ such that $X_2$ and $gX_1$ have the same distribution. In a corresponding way it is possible to introduce the concept of $G$-stability of a type of distribution. With respect to the full matrix group, only normal distributions are stable (G. Sakovich, 1960).

References

[1]	B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)

Comments

Concerning (weak) convergence of distribution functions see Distributions, convergence of; Weak convergence of probability measures.

References

[a1]	M. Loève, "Probability theory" , Princeton Univ. Press (1963)
[a2]	W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971)
[a3]	M. Sharpe, "Operator stable distributions on vector groups" Trans. Amer. Math. Soc. , 136 (1969) pp. 51–65