Functional Limit Theorem for Products of Sums of Independent and Nonidentically Distributed Random Variables

Abstract

We study the weak convergence in the space 𝐷[0,1] of processes constructed from products of sums of independent but not necessarily identically distributed random variables. The presented results extend and generalize limit theorems known so far for i.i.d. sequences.

1. Introduction

Let (𝑋_𝑛)_𝑛∈ℕ be a sequence of independent, positive, and square-integrable random variables (r.v.’s) defined on some probability space (Ω,𝔉,𝑃). For every 𝑛∈ℕ let us put 𝜇_𝑛=𝐸𝑋_𝑛,𝜏²_𝑛=Var𝑋_𝑛,𝑆_𝑛=^𝑛_𝑘=1𝑋_𝑘,𝜎²_𝑛=Var𝑆_𝑛=^𝑛_𝑘=1𝜏²_𝑘. (1) Arnold and Villaseñor [1] obtained the central limit theorem for sums of records. Rempała and Wesołowski [2] observed that this result was of general nature and proved that for a sequence of i.i.d. positive and square-integrable r.v.’s one has ∏^𝑛_𝑖=1𝑆_𝑖𝑛!𝜇^{𝑛√1/𝛾𝑛𝑑}→−𝑒^√2𝒩,as𝑛→∞, (2) where 𝒩 is a standard normal variable and 𝜇=𝜇_𝑛 and 𝛾=𝜏_𝑛/𝜇_𝑛 are the common mean and the coefficient of variation of the r.v.’s. The convergence in (2) was generalized and extended by many authors in different ways. For example, Zhang and Huang [3] studied the functional version of this result. They proved under general conditions, not involving the dependence structure of the sequence, that ∏^[𝑛𝑡]_𝑖=1𝑆_𝑖𝑛!𝜇^{𝑛√1/𝛾𝑛}_𝑑→−exp^𝑡₀𝑊(𝑥)𝑥𝑑𝑥in𝐷[0,1],as𝑛→∞, (3) here and in the sequel 𝑊(𝑥) denotes the Wiener process. The functional version of the convergence of products of sums was also studied by Kosiński [4], for i.i.d. sequences belonging to the domain of attraction of the 𝛼-stable law with 𝛼∈(1,2]. The first result for sequences of nonidentically distributed r.v.’s was obtained by Matuła and Stȩpień [5]. Similarly as in the invariance principle, the study of the non-i.i.d. sequences requires defining the processes in a different way than in (3). For 𝑡∈[0,1] and each 𝑛∈ℕ let us define the function 𝑚_𝑛(𝑡)=max{𝑖∈ℕ∪{0}∶𝜎²_𝑖≤𝑡𝜎²_𝑛}, where 𝜎²₀=0. Let us introduce the process 𝑉_𝑛(𝑡)∶=^{𝑚𝑛(𝑡)}_𝑖=1𝑆_𝑖𝐸𝑆_𝑖^{𝑎𝑖,𝑛1/𝜎𝑛} (4) and 𝑉_𝑛(𝑡)=1 for 𝑡 such that 𝑚_𝑛(𝑡)=0, where 𝑎_𝑖,𝑛=𝑓(𝜎²_𝑖/𝜎²_𝑛)(𝜏²_𝑖+1/𝜎²_𝑛)𝐸𝑆_𝑖. In [5] it was proved that for a sequence of independent, positive, and square-integrable random variables satisfying the Lindeberg condition and such that ∑^∞_𝑖=1𝜏²_𝑖+1/(𝐸𝑆_𝑖)²<∞, the process 𝑉_𝑛(𝑡) converges weakly in the space 𝐷[0,1] to the process exp(∫^𝑡₀𝑓(𝑥)𝑊(𝑥)𝑑𝑥). This result was proved in one theorem for the function 𝑓(𝑥)=1/𝑥 and in the second for functions 𝑓∈𝐶¹[0,1].

Our goal is to prove the aforementioned result for a large class of continuous functions on (0,1], which may be unbounded. Furthermore, we shall slightly change the process 𝑉_𝑛(𝑡) in order to avoid its artificial definition in the case 𝑚_𝑛(𝑡)=0 and replace the strange shifted index in 𝜏²_𝑖+1. In this setting, the imposed conditions and obtained results will be much more natural.

2. Main Results

Let ℋ be the family of functions 𝑓∶[0,1]→ℝ which are continuously differentiable on (0,1],that is,𝑓∈𝒞¹(0,1], furthermore, such that ¹₀𝑓(𝑥)1𝑥log𝑥𝑓(𝑥)𝑥𝑑𝑥<∞,_3/2→0,as𝑥→0⁺. (5)(6) Let us observe that from (5) it follows that 𝑓(𝑥)√𝑥 is integrable, if this function is also monotonic then, in consequence, (6) is satisfied. It is obvious that 𝒞¹[0,1]⊂ℋ. The family ℋ also contains functions of the form 𝑓(𝑥)=𝑥^−𝑝 with 𝑝∈(0,3/2) and 𝑓(𝑥)=1/(𝑥^3/2log^𝑟(1/𝑥∧e)) with 𝑟>3/2 and the value at 0 is arbitrary.

For any 𝑛∈ℕ let us define the process 𝑇_𝑛(𝑡)∶=^{𝑚𝑛(𝑡)+1}_𝑖=1𝑆_𝑖𝐸𝑆_𝑖^{𝛼𝑖,𝑛1/𝜎𝑛}, (7) where 𝛼_𝑖,𝑛=𝑓(𝜎²_𝑖/𝜎²_𝑛)(𝜏²_𝑖/𝜎²_𝑛)𝐸𝑆_𝑖 and we also take 𝑚_𝑛(1)=𝑛−1. In this way we obtain a sequence of processes in 𝐷[0,1] such that 𝑇_𝑛(𝑡)=^𝑘_𝑖=1𝑆_𝑖𝐸𝑆_𝑖^{𝛼𝑖,𝑛1/𝜎𝑛},𝜎for𝑡∈^2𝑘−1𝜎²_𝑛,𝜎²_𝑘𝜎²_𝑛𝑇,𝑘=1,2,…,𝑛−1,_𝑛(𝑡)=^𝑛_𝑖=1𝑆_𝑖𝐸𝑆_𝑖^{𝛼𝑖,𝑛1/𝜎𝑛},𝜎for𝑡∈^2𝑛−1𝜎²_𝑛,1, (8) recall that we have defined 𝜎²₀=0. Under our assumptions the trajectories of the process 𝑇_𝑛(𝑡) are almost surely positive functions.

Our main result concerning weak convergence of 𝑇_𝑛(𝑡) in the space 𝐷[0,1] is as follows.

Theorem 1. Let (𝑋_𝑛)_𝑛∈ℕ be a sequence of independent, positive, and square-integrable r.v.’s satisfying the Lindeberg condition: lim_𝑛→∞1𝜎²_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖−𝐸𝑋_𝑖²𝐈𝑋_𝑖−𝐸𝑋_𝑖≥𝜀𝜎_𝑛=0,∀𝜀>0, (9) and such that 𝐸𝑆_𝑛→∞,as𝑛→∞,^∞_𝑖=1𝜏²_𝑖𝐸𝑆_𝑖²<∞. (10) Then for any 𝑓∈ℋ𝑇_𝑛(𝑡)→−^𝑑exp^𝑡₀𝑓(𝑥)𝑊(𝑥)𝑑𝑥𝑖𝑛𝐷[0,1],𝑎𝑠𝑛→∞. (11)

In order to illustrate our result let us present two examples.

Example 2. Let (𝜉_𝑘)_𝑘∈ℕ be a sequence of independent r.v.’s with Poisson distribution Po(𝜆_𝑘) with parameter 𝜆_𝑘=𝑘. Let us take 𝑋_𝑘=𝜉_𝑘+1, then 𝑋_𝑘≥1 almost surely, 𝐸𝑋_𝑘=𝑘+1 and 𝜏²_𝑘=Var(𝑋_𝑘)=𝑘. Moreover, 𝐸𝑆_𝑛=2+⋯+(𝑛+1)=𝑛(𝑛+3)/2 and 𝜎²_𝑛=Var𝑆_𝑛=1+⋯+𝑛=𝑛(𝑛+1)/2. Let us observe that 𝐸|𝑋_𝑘−𝐸𝑋_𝑘|³=𝐸|𝜉_𝑘−𝐸𝜉_𝑘|³ but 𝜉_𝑘 may be considered as a sum of i.i.d. (Poisson) r.v.’s. Thus, from the well-known moment bounds, there exists a constant 𝐶>0 not depending on 𝑘 such that 𝐸|𝜉_𝑘−𝐸𝜉_𝑘|³≤𝐶⋅𝑘^3/2, for every 𝑘∈ℕ. Therefore, the Lyapunov condition may be easily verified. The condition (10) is also satisfied. We apply our Theorem 1 with 𝑓(𝑥)=1/𝑥, then 𝛼_𝑖,𝑛=(𝜏²_𝑖/𝜎²_𝑖)𝐸𝑆_𝑖=𝑖(𝑖+3)/(𝑖+1) and we have the following weak convergence in the space 𝐷[0,1], as 𝑛→∞^{𝑚𝑛(𝑡)+1}_𝑖=12𝑆_𝑖𝑖(𝑖+3)^{𝑖(𝑖+3)/(𝑖+1)1/}_𝑑^{𝑛(𝑛+1)/2}→−exp^𝑡₀𝑊(𝑥)𝑥𝑑𝑥. (12)

It may be easily checked (see page  42 in [6] for details) that the integral ∫¹₀𝑥^𝑝𝑊(𝑥)𝑑𝑥 has Gaussian distribution 𝒩(0,𝜎²) with 𝜎²=¹₀min𝑥,𝑦𝑥^𝑝𝑦^𝑝2𝑑𝑥𝑑𝑦=,𝑝+22𝑝+3 (13) provided 𝑝>−3/2. Therefore, we have the following example.

Example 3. Let (𝑋_𝑘)_𝑘∈ℕ be a sequence of i.i.d. r.v.’s with the standard exponential distribution. Then 𝐸𝑋_𝑘=Var(𝑋_𝑘)=1 and 𝜎²_𝑛=𝑛. We apply Theorem 1 with 𝑡=1 and the function 𝑓(𝑥)=𝑥^𝑝, where 𝑝>−3/2. In this case 𝛼_𝑖,𝑛=(𝑖/𝑛)^𝑝+1 and we get ^𝑛_𝑖=1𝑆_𝑖𝑖^{𝑖𝑝+11/𝑛(𝑝+3/2)𝑑}→−exp2𝒩,𝑝+22𝑝+3as𝑛→∞. (14)

3. Proofs and Auxiliary Results

In the proof of our main result, we shall use the following lemma.

Lemma 4. Let 𝑑_𝐷[0,1] be the metric making the space 𝐷[0,1] to be a complete and separable metric space (Polish space) and let 𝑑_𝑈(𝑥(𝑡),𝑦(𝑡))=sup_0≤𝑡≤1|𝑥(𝑡)−𝑦(𝑡)| be the uniform metric. Furthermore, denote by 𝑑_{𝐿𝑃𝐷[0,1]} the Lévy-Prohorov metric in the space of probability measures on 𝐷[0,1] and by 𝑑_{𝐾𝐹𝐷[0,1]} the Ky-Fan metric. Let 𝑋, 𝑌, and 𝑋_𝑛 be random elements with values in 𝐷[0,1], then (a)if 𝐸𝑑_𝑈(𝑋,𝑌)≤𝑎, then 𝑑_{𝐾𝐹𝐷[0,1]}(𝑋,𝑌)≤√𝑎,(b)𝑑_{𝐿𝑃𝐷[0,1]}(𝑋,𝑌)≤𝑑_{𝐾𝐹𝐷[0,1]}(𝑋,𝑌), (c)if 𝑑_𝑈(𝑋_𝑛,𝑋)→0 almost surely, then 𝑑_{𝐾𝐹𝐷[0,1]}(𝑋_𝑛,𝑋)→0.

Proof. The proof is based on the inequality 𝑑_𝐷[0,1]≤𝑑_𝑈, which may be found on page  150 in [7]. The inequality (b) is proved, for example, in Theorem 11.3.5 on page  397 in [8].

Proof of Theorem 1. In the first step we shall prove that log𝑇_𝑛(𝑡)=1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑛𝑆_𝑖−𝐸𝑆_𝑖+𝐴_𝑛(𝑡), (15) where 𝐴−→_𝑛^𝑃0 in 𝐷[0,1], as 𝑛→∞. We shall apply the expansion of the logarithm log(1+𝑥)=𝑥+𝑅(𝑥), for |𝑥|≤1/2, where |𝑅(𝑥)|≤2𝑥². Let us put 𝐶_𝑖=(𝑆_𝑖−𝐸𝑆_𝑖)/𝐸𝑆_𝑖, then we easily get 𝑆log^𝑖𝐸𝑆_𝑖=log𝐶_𝑖+1=𝐶_𝑖+𝑅𝐶_𝑖𝐈𝐶_𝑖≤12+log1+𝐶_𝑖−𝐶_𝑖×𝐈𝐶_𝑖>12,log𝑇_𝑛(𝑡)=1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑛𝑆log^𝑖𝐸𝑆_𝑖=1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑛𝐶_𝑖+1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑛𝑅𝐶_𝑖𝐈𝐶_𝑖≤12+1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑛log1+𝐶_𝑖−𝐶_𝑖×𝐈𝐶_𝑖>12=1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑛𝑆_𝑖−𝐸𝑆_𝑖+𝐴^′_𝑛(𝑡)+𝐴_𝑛^′′(𝑡),say. (16) Let us observe that 𝐸max_0≤𝑡≤1𝐴^′_𝑛(𝑡1)≤𝐸𝜎_𝑛^𝑛_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑛𝑅𝐶_𝑖𝐈𝐶_𝑖≤12≤2𝜎_𝑛^𝑛_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑛𝐸𝑆_𝑖−𝐸𝑆_𝑖𝐸𝑆_𝑖²=2^𝑛_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜎_𝑖𝜎_𝑛𝜏²_𝑖𝜎²_𝑛𝜎_𝑖𝐸𝑆_𝑖. (17) From the assumption (10) and the Kronecker lemma it follows that (1/(𝐸𝑆_𝑛)²)∑^𝑛_𝑖=1𝜏²_𝑖=𝜎²_𝑛/(𝐸𝑆_𝑛)²→0. It is easy to see that from the Lindeberg condition (9), the Feller condition lim_𝑛→∞max_{1≤𝑖≤𝑛}𝜏²_𝑖𝜎²_𝑛=0 (18) holds; therefore, ^𝑛_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜎_𝑖𝜎_𝑛𝜏²_𝑖𝜎²_𝑛→¹₀𝑓(𝑥)√𝑥𝑑𝑥<∞. (19) Moreover, by our assumption (6) we get 𝑓𝜎²_𝑖𝜎²_𝑛𝜎_𝑖𝜎_𝑛𝜏²_𝑖𝜎²_𝑛≤𝑓𝜎²_𝑖𝜎²_𝑛𝜎_𝑖𝜎_𝑛³→0. (20) By the above remarks and Toeplitz theorem on the transformation of sequences into sequences (Problem 2.3.1 in [9]) we obtain ^𝑛_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜎_𝑖𝜎_𝑛𝜏²_𝑖𝜎²_𝑛𝜎_𝑖𝐸𝑆_𝑖→0. (21) Thus from (17) we have 𝐴−→^′_𝑛^𝑃0 in 𝐷[0,1], as 𝑛→∞.
From (10) the strong law of large numbers holds, that is, 𝐶_𝑛=(𝑆_𝑛−𝐸𝑆_𝑛)/𝐸𝑆_𝑛→0 almost surely. Thus, for almost all 𝜔∈Ω and sufficiently large 𝑖≥𝑖₀(𝜔) there holds 𝐈[|𝐶_𝑖(𝜔)|>1/2]=0. Since that max_0≤𝑡≤1𝐴_𝑛^′′(𝑡≤1)𝜎_𝑛^𝑛_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝐸𝑆_𝑖𝜎²_𝑖log1+𝐶_𝑖−𝐶_𝑖×𝐈𝐶_𝑖>12→0,a.s. (22) Consequently 𝐴_𝑛^𝑃′′−→0 in 𝐷[0,1], and (15) is proved.
It remains to prove that 𝑌_𝑛(𝑡)1∶=𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑛𝑆_𝑖−𝐸𝑆_𝑖𝑑→−^𝑡₀𝑓(𝑥)𝑊(𝑥)𝑑𝑥 (23) in 𝐷[0,1], as 𝑛→∞.
Let 0<𝜀<1 be given and let us define a process 𝑌_𝑛,𝜀(𝑡)=𝜎0,0≤𝑡<^{2𝑚𝑛(𝜀)}𝜎²_𝑛,1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_𝑖=𝑚∑_𝑛(𝜀)+1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑛𝑆_𝑖−𝐸𝑆_𝑖,𝜎²_{𝑚𝑛(𝜀)}𝜎²_𝑛≤𝑡≤1. (24)
Let us observe that 𝐸max_0≤𝑡≤1𝑌_𝑛(𝑡)−𝑌_𝑛,𝜀(𝑡1)≤𝐸𝜎_𝑛^{𝑚𝑛(𝜀)}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑖𝑆_𝑖−𝐸𝑆_𝑖≤^{𝑚𝑛(𝜀)}_𝑖=1𝑓𝜎²_𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑛𝜎_𝑖𝜎_𝑛→^𝜀₀𝑓(𝑥)√𝑥𝑑𝑥,as𝑛→∞,forfixed𝜀>0. (25)
Since from the definition of 𝑚_𝑛(𝑡) we have 𝜎²_{𝑚𝑛(𝜀)}/𝜎²_𝑛→𝜀, therefore, by the integrability of the function 𝑓(𝑥)√𝑥 and Lemma 4(a) we get lim_𝜀→0limsup_𝑛→∞𝑑_KF𝐷[0,1]𝑌_𝑛(𝑡),𝑌_𝑛,𝜀(𝑡)=0. (26)
Let us define a continuous mapping 𝐻_𝜀∶𝐷[0,1]→𝐶[0,1]𝐻_𝜀𝑔(𝑡)=∫0,0≤𝑡<𝜀,_𝑡𝜀𝑔(𝑥)𝑓(𝑥)𝑑𝑥,𝜀<𝑡≤1. (27) The above integral is properly defined since 𝑔 is bounded and 𝑓∈𝒞¹[𝜀,1]. In the next step we shall prove that 𝐻_𝜀𝑊_𝑛(𝑡)=𝑌_𝑛,𝜀(𝑡)+𝐵_𝑛(𝑡), (28) where 𝐵−→_𝑛^𝑃0 in 𝐷[0,1], as 𝑛→∞ and 𝑊_𝑛(𝑡)=^{𝑚𝑛(𝑡)+1}_𝑘=1𝑋_𝑘−𝜇_𝑘𝜎_𝑛. (29)
At first let us observe that 𝐸max_{0≤𝑡≤𝜀}𝐻_𝜀𝑊_𝑛(𝑡)−𝑌_𝑛,𝜀(𝑡=1)𝜎_𝑛𝑓𝜎²_{𝑚𝑛(𝜀)+1}𝜎²_𝑛𝜏²_{𝑚𝑛(𝜀)+1}𝜎²_𝑛𝐸𝑆_{𝑚𝑛(𝜀)+1}−𝐸𝑆_{𝑚𝑛(𝜀)+1}≤𝜎_{𝑚𝑛(𝜀)+1}𝜎_𝑛sup_{𝜀≤𝑥≤1}𝑓(𝑥)max_{1≤𝑖≤𝑛}𝜏²_𝑖𝜎²_𝑛→0,as𝑛→∞, (30) by the Feller condition (18).
In the case 𝜀<𝑡≤1, let us calculate the integral 𝐻_𝜀𝑊_𝑛(𝑡)=^𝑡_𝜀𝑓(𝑥)𝑊_𝑛(𝑥)𝑑𝑥=−^𝜀_{𝜎2𝑚𝑛(𝜀)/𝜎2𝑛}𝑓(𝑥)𝑊_𝑛(𝑥)−𝑑𝑥_{𝜎2𝑚𝑛(𝑡)+1/𝜎2𝑛𝑡}𝑓(𝑥)𝑊_𝑛(𝑥)+𝑑𝑥_{𝑚𝑛(𝑡)+1𝑖=𝑚𝑛(𝜀)+1𝜎2𝑖/𝜎2𝑛𝜎2𝑖−1/𝜎2𝑛}𝑓(𝑥)𝑊_𝑛(𝑥)𝑑𝑥=𝑇^′_𝑛(𝑡)+𝑇_𝑛^′′(𝑡)+𝑇_𝑛^′′′(𝑡),say. (31)
In fact 𝑇^′_𝑛(𝑡) does not depend on 𝑡, thus 𝐸max_{𝜀≤𝑡≤1}𝑇^′_𝑛(𝑡−)=𝐸^𝜀_{𝜎2𝑚𝑛(𝜀)/𝜎2𝑛}𝑓(𝑥)𝑊_𝑛(𝑥)≤𝜎𝑑𝑥𝜀−^{2𝑚𝑛(𝜀)}𝜎²_𝑛×sup_{𝜎2𝑚𝑛(𝜀)/𝜎2𝑛≤𝑥≤1}𝑓(𝑥𝐸𝑆)^{𝑚𝑛(𝜀)+1}−𝐸𝑆_{𝑚𝑛(𝜀)+1}𝜎_𝑛→0, (32) as 𝑛→∞, since 𝜎²_{𝑚𝑛(𝜀)}/𝜎²_𝑛→𝜀 and 𝐸|𝑆_{𝑚𝑛(𝜀)+1}−𝐸𝑆_{𝑚𝑛(𝜀)+1}|≤𝜎_{𝑚𝑛(𝜀)+1}≤𝜎_𝑛.
Let us find the bounds for the second term 𝑇_𝑛^′′(𝑡): 𝑇_𝑛^′′(𝑡)=^{𝜎2𝑚𝑛(𝑡)+1/𝜎2𝑛}_𝑡𝑓(𝑥)𝑊_𝑛(𝑥)≤𝑑𝑥max_{1≤𝑖≤𝑛}𝜏²_𝑖𝜎²_𝑛sup_{𝜀≤𝑥≤1}𝑓(𝑥𝑆)^{𝑚𝑛(𝑡)+1}−𝐸𝑆_{𝑚𝑛(𝑡)+1}𝜎_𝑛. (33) Further, for any 𝛿>0, we have 𝑃max_{𝜀≤𝑡≤1}𝑇_𝑛^′′(𝑡)>𝛿≤𝑃max𝜏²_𝑖𝜎³_𝑛sup_{𝜀≤𝑥≤1}𝑓(𝑥)max_{𝜀≤𝑡≤1}𝑆_{𝑚𝑛(𝑡)+1}−𝐸𝑆_{𝑚𝑛(𝑡)+1}>𝛿≤𝑃max𝜏²_𝑖𝜎³_𝑛sup_{𝜀≤𝑥≤1}𝑓(𝑥)max_{1≤𝑖≤𝑛}𝑆_𝑖−𝐸𝑆_𝑖≤>𝛿max𝜏²_𝑖𝜎²_𝑛²sup_{𝜀≤𝑥≤1}𝑓(𝑥)²1𝛿²→0,as𝑛→∞, (34) by the Kolmogorov inequality and the Feller condition (18).
Finally, let 𝑛 be sufficiently large, that is, such that 𝜎²_{𝑚𝑛(𝜀)}/𝜎²_𝑛≥𝜀/2, then max_{𝜀≤𝑡≤1}𝑇_𝑛^′′′(𝑡)−𝑌_𝑛,𝜀(𝑡)≤max_{𝜀≤𝑡≤1}1𝜎_𝑛^{𝑚𝑛(𝑡)+1}_{𝑖=𝑚𝑛(𝜀)+1}𝑆_𝑖−𝐸𝑆_𝑖×^{𝜎2𝑖/𝜎2𝑛}_{𝜎2𝑖−1/𝜎2𝑛}𝑓(𝑥)𝜎𝑑𝑥−𝑓^2𝑖𝜎²_𝑛𝜏²_𝑖𝜎²_𝑛≤max_{1≤𝑖≤𝑛}𝑆_𝑖−𝐸𝑆_𝑖𝜎_𝑛×max^𝑚_{𝜀≤𝑡≤1}^𝑛(𝑡)+1_{𝑖=𝑚𝑛(𝜀)+1}^{𝜎2𝑖/𝜎2𝑛}_{𝜎2𝑖−1/𝜎2𝑛}𝑓(𝑥)𝜎−𝑓^2𝑖𝜎²_𝑛≤𝑑𝑥max_{1≤𝑖≤𝑛}𝑆_𝑖−𝐸𝑆_𝑖𝜎_𝑛^𝑛_𝑖=1𝑐_𝑓,𝜀𝜏²_𝑖𝜎²_𝑛²≤max_{1≤𝑖≤𝑛}𝑆_𝑖−𝐸𝑆_𝑖𝜎_𝑛max𝜏²_𝑖𝜎²_𝑛𝑐_𝑓,𝜀, (35) where 𝑐_𝑓,𝜀=max_{𝜀/2≤𝑥≤1}|𝑓^′(𝑥)|. Thus, again by the Kolmogorov inequality and the Feller condition, we get 𝑃max_{𝜀≤𝑡≤1}𝑇_𝑛^′′′(𝑡)−𝑌_𝑛,𝜀(𝑡)>𝛿≤𝑃max_{1≤𝑖≤𝑛}𝑆_𝑖−𝐸𝑆_𝑖𝜎_𝑛max𝜏²_𝑖𝜎²_𝑛𝑐_𝑓,𝜀≤>𝛿max𝜏²_𝑖𝜎²_𝑛𝛿𝑐_𝑓,𝜀²→0,as𝑛→∞. (36)
Consequently, from (30), (32), (34), and (36) the statement (28) follows.
To end the proof, let us note that for the process 𝑊_𝑛(𝑡) defined by (29), we have 𝑊_𝑛(𝑡)→−^𝑑𝑊(𝑡) in 𝐷[0,1], by the result of Prohorov (see Problem 1, page  77 in [7]). In fact, Prohorov considered a broken-line process 𝑋_𝑛(𝑡) and its convergence in 𝐶[0,1], but sup_0≤𝑡≤1𝑊_𝑛(𝑡)−𝑋_𝑛(𝑡)≤max_{1≤𝑖≤𝑛}𝑋_𝑖−𝐸𝑋_𝑖𝜎_𝑛^𝑃−→0, (37) from the Lindeberg condition (9). Thus, by the continuity of 𝐻_𝜀, we get 𝐻_𝜀𝑊_𝑛(𝑡)→−^𝑑𝐻_𝜀(𝑊)(𝑡),in𝐷[0,1],as𝑛→∞. (38)
Furthermore, sup_0≤𝑡≤1𝐻_𝜀(𝑊)(𝑡)−^𝑡₀𝑓(𝑥)𝑊(𝑥)𝑑𝑥=sup_{0≤𝑡≤𝜀}^𝑡₀𝑓(𝑥)𝑊(𝑥)𝑑𝑥≤sup_{0<𝑥≤𝜀}|𝑊(𝑥)|𝑥log(1/𝑥)⋅^𝜀₀𝑓(𝑥)1𝑥log𝑥𝑑𝑥→0,almostsurelyas𝜀→0⁺, (39) by (5) and the properties of the modulus of continuity of the Wiener process, that is, limsup_𝜀→0+sup_{0<𝑥≤𝜀}|𝑊(𝑥)|()≤√𝑥log1/𝑥2almostsurely; (40) see formula (7) on page  534 in [6].
For short, let us put 𝐻(𝑊)(𝑡)=∫^𝑡₀𝑓(𝑥)𝑊(𝑥)𝑑𝑥. Then, with the notation introduced in Lemma 4, we get 𝑑_LP𝐷[0,1]𝑌_𝑛;𝐻(𝑊)≤𝑑_LP𝐷[0,1]𝑌_𝑛;𝑌_𝑛,𝜀+𝑑_LP𝐷[0,1]𝑌_𝑛,𝜀;𝐻_𝜀𝑊_𝑛+𝑑_LP𝐷[0,1]𝐻_𝜀𝑊_𝑛;𝐻_𝜀(𝑊)+𝑑_LP𝐷[0,1]𝐻_𝜀(𝑊);𝐻(𝑊)≤𝑑_KF𝐷[0,1]𝑌_𝑛;𝑌_𝑛,𝜀+𝑑_KF𝐷[0,1]𝑌_𝑛,𝜀;𝐻_𝜀𝑊_𝑛+𝑑_LP𝐷[0,1]𝐻_𝜀𝑊_𝑛;𝐻_𝜀(𝑊)+𝑑_KF𝐷[0,1]𝐻_𝜀(𝑊);𝐻(𝑊). (41) Now, (23) follows from (26), (28), (38), and (39) and the proof is completed.