On the transfer theorems for observed and
unobserved random variables
Abstract:
We characterize the possible weak limits of \[\sum _{i=1}^n \epsilon _iX_i/k_n\]
for a sequence \(\{X_n, n\geq 1\}\) of independent random variables
and a sequence \(\{\epsilon _n, n\geq
1\}\) of indicator random variables (\(P[\epsilon _n\in\{0,1\}]=1\)
for \(n\geq 1\)) and a non-random normalizing sequence \(\{k_n, n\geq 1\}\)
of positive reals. We consider two cases: when \(\{X_n, n\geq 1\}\) and \(\{\epsilon _n,
n\geq 1\}\) are independent or dependent.
In the first case we obtain results generalizing transfer theorems, whereas in the other case, only a partial characterization was possible.
2010 AMS Mathematics Subject Classification: Primary 60F05.
Keywords and phrases: transfer theorem, weak limits, sums of observed random
variables.
|
B. V. Gnedenko and H. Fahim, On a transfer theorem, Dokl. Akad. Nauk SSSR 187 (1969), 15-17 (in Russian).
P. Kern, A general multiparameter version of Gnedenko's transfer theorem, Teor. Veroyatn. Primenen. 60 (2015), 198-206 (in Russian);
English transl.: Theory Probab. Appl. 60 (2016), 134-142.
L. B. Klebanov and S. T. Rachev, Sums of random variables and their approximations with ν -accompanying
infinitely divisible laws, Serdica Math. J. 22 (1996), 471-496.
T. Krajka and Z. Rychlik, The limiting behaviour of sums and maximums of iid random variables from the viewpoint of different observers, Probab. Math. Statist. 34 (2014), 237-252.
V. M. Kruglov and V. Yu. Korolev, Limit Theorems for Random Sums, Moscow Univ. Press, Moscow, 1990 (in Russian).
A. Rényi, Probability Theory, Akadémiai Kiadó, Budapest, 1970.
D. Szász, On classes of limit distributions for sums of a random number of identically distributed random variables, Teor. Veroyatn. Primenen. 17 (1972), 424-439 (in Russian); English transl.: Theory Probab. Appl. 17 (1972), 401–415.
D. Szász, Limit theorems for the distributions of the sums of a random number of random variables, Ann. Math. Statist. 43 (1972), 1902-1913.
D. Szász and B. Freyer, One problem of summation with a random index, Lithuanian Math. J. 11 (1971), 181-187.
|
